A Triple-Error-Correcting Cyclic Code from the Gold and Kasami-Welch APN Power Functions
Based on a sufficient condition proposed by Hollmann and Xiang for constructing triple-error-correcting codes, the minimum distance of a binary cyclic code $\mathcal{C}_{1,3,13}$ with three zeros $\alpha$, $\alpha^3$, and $\alpha^{13}$ of length $2^m…
Authors: Xiangyong Zeng, Jinyong Shan, Lei Hu
A T riple-Erro r-Correc ting Cyclic Co d e from the Gold and Kasami-W elc h APN P o w er F uncti ons Xiangy ong Zeng, Jin y ong Shan, Le i Hu ∗ † Octob er 17, 2018 Abstract: Based on a sufficient co nd ition pr op osed b y Hollmann and Xiang for constructing triple-error-correcting co des, the minim um distance of a binary cyclic co de C 1 , 3 , 13 with three zeros α , α 3 , and α 13 of length 2 m − 1 and the w eigh t d ivisibilit y of i ts dual code are studied, where m ≥ 5 is o d d and α is a p rimitiv e elemen t of the finite field F 2 m . T h e co de C 1 , 3 , 13 is pro ve n to ha v e the same weig ht distribution as the binary triple-error-correcting primitive BCH co d e C 1 , 3 , 5 of the same length. Keyw ords: Cyclic c o d e, BCH co de, trip le-error-correcting co de, min im um distance, almost p erfect nonlinear function 1 In tro du ction In co d ing th eory , bin ary triple-error-correcting p rimitiv e BCH cod es of length n = 2 m − 1 are one of the most studied ob j ects [6, 15]. Le t α b e a primitiv e elemen t of th e fi nite field F 2 m with 2 m elemen ts, and for a subset I of Z 2 m − 1 , let C I denote the length- n cyclic co de with zeros α i ( i ∈ I ). T he primitiv e BCH co de C 1 , 3 , 5 has minimum distance 7, and its weigh t d istribution w as discussed in [19, 1 , 2, 3]. F or some other int egers d 1 and d 2 (they are n aturally assumed to b e differen t in the sens e of cycloto mic equiv alence mo dulo 2 m − 1 and b e different to 1), the co de C 1 ,d 1 ,d 2 can also ha v e th e same w eigh t d istribution as the binary triple-error-correcting primitive BCH code C 1 , 3 , 5 . F or example, T able 1 lists all kn own such exp onen t pairs { d 1 , d 2 } f or od d ∗ X. Zeng and J. Shan a re with F acult y of Mathematics and Computer Science, Hub ei Un ivers ity , W uhan 430062, China ( e-mail: xiangyongzeng@y aho o.com.cn). † L. Hu is with th e S tate Key Lab oratory of I nformation Securit y , Graduate School of Chinese Academ y of Sciences, Beij ing 100049, China (e-mail: hu@is.ac.cn). 1 m , where there exists only one class of exp onents with binary w eigh t greater than 2, namely (2 m +1 2 + 1) 2 in the construction of [10]. T able 1: Kno wn exp onent pairs { d 1 , d 2 } for o dd m such that C 1 ,d 1 ,d 2 and C 1 , 3 , 5 ha ve the same w eigh t d istributions d 1 , d 2 condition 2 r + 1 , 2 2 r + 1 m o d d , gcd( m, r ) = 1 [20] 2 r + 1 , 2 3 r + 1 m o d d , gcd( m, r ) = 1 [20] 2 m − 1 2 + 1 , 2 m − 1 2 − 1 + 1 m o d d [22] 2 m +1 2 + 1 , (2 m +1 2 + 1) 2 m o d d [10] Recen tly , Hollmann and Xiang [16] prop osed a s u fficien t condition for constru cting binary triple-error-correcting co des of length n = 2 m − 1 for o dd m . More precisely , if a b inary cyclic co de C of length n = 2 m − 1 and d imension n − 3 m has minim u m d istance at least 7, and if the weig hts of all cod ew ords of its dual co de C ⊥ are divisible b y 2 m − 1 2 , then C has the same w eigh t d istribution as the co d e C 1 , 3 , 5 . F or t w o exp onen ts d 1 and d 2 suc h that b oth x d 1 and x d 2 are almost p erfect nonlinear (APN) p o w er functions from F 2 m to itself, eac h of the co des C 1 ,d 1 and C 1 ,d 2 has min im um distance exactly 5 b y Theorem 5 of [9] (see also Lemma 1 in Section 2). Notice that C 1 ,d 1 ,d 2 is a sub co de of b oth C 1 ,d 1 and C 1 ,d 2 , then C 1 ,d 1 ,d 2 has min imum distance at least 5. This motiv ate s us to lo ok for suitable APN p o wer exp onen ts d 1 and d 2 suc h that C 1 ,d 1 ,d 2 has the same weig ht distribution as C 1 , 3 , 5 . T able 2: Kno wn v alues of APN p o w er exp onen ts for o dd m T yp e d condition Gold 2 r + 1 gcd( r , m ) = 1 [14] Kasami-W elc h 2 2 r − 2 r + 1 gcd( r , m ) = 1 [20] W elc h 2 m − 1 2 + 3 m o d d [24] Niho 2 2 r + 2 r − 1 4 r ≡ − 1 (mo d m ), m o dd [24] In verse 2 m − 1 − 1 m o d d [4, 25] Dobb ertin 2 4 r + 2 3 r + 2 2 r + 2 r − 1 m = 5 r , m o d d [13] F ollo wing this idea, we exp erimen tally test all known v alues of APN p ow er exp onen ts (listed in T able 2) for o d d intege r s m = 5, 7, 9 and 11, to try to fin d pairs ( d 1 , d 2 ) suc h th at C 1 ,d 1 ,d 2 and C 1 , 3 , 5 ha ve the same weigh t distributions. By the MacWilliams identit y for bin ary linear co des [22], this is equiv alen t to say that their dual co des C ⊥ 1 ,d 1 ,d 2 and C ⊥ 1 , 3 , 5 ha ve the same weigh t 2 distributions. The w eigh t d istribution of C ⊥ 1 , 3 , 5 is give n in [19, 22]. The dual co d e C ⊥ 1 ,d 1 ,d 2 is simply given by C ⊥ 1 ,d 1 ,d 2 = n c ( ǫ, γ , δ ) = T r m 1 ( ǫx + γ x d 1 + δ x d 2 ) x ∈ F ∗ 2 m | ǫ, γ , δ ∈ F 2 m o (1) and its we ight distribu tion is b etter to compute th an that of the target co d e C 1 ,d 1 ,d 2 . All APN exp onen t pairs ( d 1 , d 2 ) suc h that C ⊥ 1 ,d 1 ,d 2 and C ⊥ 1 , 3 , 5 ha ve the same we ight distri- butions in our exp eriment are listed in T a b le 3. F or o d d m and gcd( r , m ) = 1, the co de C 2 r +1 , 2 3 r +1 , 2 5 r +1 also has the same weig ht distribution as C 1 , 3 , 5 [20]. T his construction and th ose in T a b le 1 can explain all pairs { d 1 , d 2 } without th e m ark ⋆ in T able 3. Noti ce that we sa y a p air ( d 1 , d 2 ) h as actually b een explained if C d, 2 i 1 d 1 d, 2 i 2 d 2 d is pro ve n to h a v e th e same wei ght distribution as C 1 , 3 , 5 for thr ee integ ers i 1 , i 2 , d with 0 ≤ i 1 , i 2 ≤ m − 1, gcd( d, 2 m − 1) = 1 sin ce C 1 ,d 1 ,d 2 and C d, 2 i 1 d 1 d, 2 i 2 d 2 d ha ve the same w eigh t distributions, where the s ubscripts are tak en mo dulo 2 m − 1. T able 3: Exp onent pairs ( d 1 , d 2 ) suc h th at C 1 ,d 1 ,d 2 and C 1 , 3 , 5 ha ve the same w eigh t distributions for m = 5, 7, 9 and 11 Exp onent p air ( d 1 , d 2 ) m = 5 m = 7 m = 9 m = 11 (Gold, Gold) (3,5) (3,5), (3,9) (3,5), (3,9) (3,5),( 3,9),(3,17),(3,33) (5,9) (3,17) ,(5,9) (5,9),(5,17 ),(5,33),(9,17) (5,17) ,(9,17) (9,33) , (17,3 3) (Gold, Kasami-W elc h) (3 , 13 ) (3 , 13) ⋆ ,(9,13) (3 , 13) ⋆ (3 , 13) ⋆ (Gold, W elc h) (5,7) (3,11) ,(5 , 11) ⋆ (Gold, Niho) (3,5) (Kasami-W elc h, W elc h) (13 , 7) (Kasami-W elc h, Niho) (13 , 39 ) Indeed, w e fi nd a new pair marke d b y ⋆ whic h can not b e exp lained b y kno wn results, where we regard (5 , 11) and (3 , 13) as a same pair since C 1 , 5 , 11 has the same weig ht distribution as C 13 , 2 × 5 × 13 , 2 3 × 11 × 13 , i.e., C 1 , 3 , 13 . It is the Gold exp onen t d 1 = 3 and Kasami-W elc h exp onent d 2 = 13 , and the latter is another example of exp on ents with b inary weigh t 3. This pap er w ill pro ve th at for any od d in teger m ≥ 5, the co d e C 1 , 3 , 13 has the same weig ht distribution as C 1 , 3 , 5 . T o this end, w e use a method dev elop ed b y Hol lmann and Xiang in [16, 17] whic h analyzes the divisibilit y of th e weigh ts of the codewords in C ⊥ 1 , 3 , 13 b y an add -with-carry algorithm and a tec hnical graph-theoretic deduction. In reference [16], Hollmann and Xiang also applied this m etho d to stud y the co de C 1 ,d 1 ,d 2 prop osed in [10], where d 1 = 2 m +1 2 + 1 and 3 d 2 = (2 m +1 2 + 1) 2 are d ep endent on m . Th e pair (3 , 13) in this pap er is indep enden t on m , and this makes the divisibilit y analysis more complex than that in [16]. The r emainder of this pap er is organized as follo ws. Section 2 giv es some preliminaries and the results of this pap er. Section 3 establishes a low er b ound on the minimum distance of the co de C 1 , 3 , 13 . Section 4 discusses th e we ight d ivisibilit y of C ⊥ 1 , 3 , 13 . Section 5 concludes th e stu dy . 2 Preliminaries and the Results Let F ∗ 2 m = F 2 m \ { 0 } . The trace fun ction T r m 1 from F 2 m to F 2 is defined by [21] T r m 1 ( x ) = m − 1 X i =0 x 2 i , x ∈ F 2 m . A binary cyclic co de C of length n is a p rincipal ideal in the r in g F 2 [ x ] / ( x n − 1). If g ( x ) is a generator p olynomial of C , then a p o w er β of a primitiv e n -th ro ot of un it y is a zero of th e co de C if and only if g ( β ) = 0. A co dew ord c in C has the form as c 0 + c 1 x + · · · + c n − 1 x n − 1 , which corresp onds to a b inary v ector ( c 0 , c 1 , · · · , c n − 1 ). T he Hamming weigh t of the co dew ord c is the n umb er of nonzero c i for 0 ≤ i ≤ n − 1, den oted by w t ( c ). Definition 1: A f unction f fr om F 2 m to itself is said to b e almost p erfect nonlinear (APN) if for eac h e ∈ F ∗ 2 m , th e fun ction ∆ f , e ( x ) = f ( x + e ) + f ( x ) is tw o-to- one from F 2 m to itself. APN functions were in tro d uced in [25] b y Nyb erg to defin e them as th e mappin gs with highest resistance to differen tial cryp tanalysis. F or more details we r efer the reader to [4, 7, 8, 11, 12, 13, 14, 18, 20, 25] and the references therein. F or a function f from F 2 m to itself with f (0) = 0, let C f denote the binary cyclic cod e of length n = 2 m − 1 with parit y c hec k matrix H f = 1 α α 2 · · · α 2 n − 2 f (1) f ( α ) f ( α 2 ) · · · f ( α 2 n − 2 ) ! where eac h en try is view ed as a binary column vec tor basing on a basis expression of elemen ts of F 2 m o v er F 2 . The APN prop erties of f can b e c haracterized by the min im um distance of C f [9]. Lemma 1: ([9]) The co de C f has min imum distance 5 if and only if f is APN. Since the 1960s, the family of triple-error-correcting bin ary primitive BCH co des of length n = 2 m − 1 has b een thoroughly stud ied. T h e follo wing lemma giv en by Hollmann and Xiang present ed a su fficien t condition f or constructing families of triple-error-correcting co d es. 4 Lemma 2: ([16]) Let m b e o dd and C b e a binary cyclic co de of length n = 2 m − 1, dimension n − 3 m and minimum distance at least 7. If all w eights of the cod ew ords in C ⊥ are divisible b y 2 m − 1 2 , th en C has the same weig ht distribution as C 1 , 3 , 5 . With Lemma 2, for o dd m , we can construct bin ary triple-error-correcting co d es of length n = 2 m − 1 and d im en sion n − 3 m by analyzing their minimum distances and weigh t d ivisibilit y of their d ual co des. The follo wing Pr op osition 1 will b e p ro ve n in the next section, and the follo wing Lemma 3 sho ws that the pro duct of the nonzeros of a binary cyclic co de can b e used to analyze the weigh t divisibilit y . Prop osition 1: F or o dd m ≥ 5, the co de C 1 , 3 , 13 has min imum distance at least 7. Lemma 3: ([23]) Let C b e a b inary cyclic co de, and let l b e the smallest p ositiv e integ er suc h that l n onzeros of C (with rep etitio n s allo w ed) ha v e p ro duct 1. Th en the weigh t of ev ery co dew ord in C is divisible b y 2 l − 1 , and there is at least one co d ew ord wh ose weig ht is not divisible b y 2 l . Based on L emma 3, Hollmann and Xiang presented an add-with-carry algorithm to obtain information on the largest p o w er of 2 d ividing the weig hts of all co dewo r d s of a binary cyclic co de as b elo w [16, 17]. F or a p ositiv e int eger m and a n on-negativ e in teger a with the b inary expression a = m − 1 P i =0 a i 2 i , a i ∈ { 0 , 1 } , the (binary) we ight w ( a ) of a is d efi ned as the inte ger w ( a ) = m − 1 P i =0 a i . F or d 1 , d 2 , · · · , d j ∈ Z 2 m − 1 , defin e M ( m ; d 1 , d 2 , · · · , d j ) = max w ( s ) − j X l =1 w ( a ( l ) ) ! where th e maxim um is taken ov er all inte gers s , a (1) , · · · , a ( j ) satisfying 0 ≤ s, a (1) , · · · , a ( j ) ≤ 2 m − 1 , s ≡ j X l =1 d l a ( l ) (mo d 2 m − 1) and a ( l ) 6≡ 0 (mod 2 m − 1) for s ome l . The add-with-carry algorithm for in tegers mo dulo 2 m − 1 can b e u sed to determine M ( m ; d 1 , d 2 , · · · , d j ) [16, 17]. Let a ( l ) and s ha ve bin ary expressions a ( l ) = m − 1 X i =0 a ( l ) i 2 i for 1 ≤ l ≤ j and s = m − 1 X i =0 s i 2 i , (2) resp ectiv ely . F urthermore, let d 1 , d 2 , · · · , d j b e nonzero inte gers, and d efine d + = P d l > 0 d l and 5 d − = P d l < 0 d l so that j P l =1 d l = d + + d − , d + ≥ 0 , d − ≤ 0 , and supp ose that s ≡ d 1 a (1) + d 2 a (2) + · · · + d j a ( j ) (mo d 2 m − 1) . Lemma 4: ([16, 17]) There exists a un ique int eger sequence c − 1 , c 0 , . . . , c m − 1 with c − 1 = c m − 1 suc h that 2 c i + s i = j X l =1 d l a ( l ) i + c i − 1 , 0 ≤ i ≤ m − 1 (3) holds. Moreo v er, w ith notation w ( c ) = m − 1 P i =0 c i , we ha ve that w ( c ) = j X l =1 d l w ( a ( l ) ) − w ( s ) . The num b ers c i satisfy d − − 1 ≤ c i ≤ d + , and further d − ≤ c i < d + for all i if a ( l ) 6≡ 0 (mod 2 m − 1) holds for some l . The integers s i and c i are calle d the digits and c arries for the computation of s mo du lo 2 m − 1 in terms of a (1) , · · · , a ( j ) , d 1 , · · · , d j . Lemma 5: ([16, 17]) All the weig hts of C ⊥ 1 ,d 1 ,d 2 are divisible by 2 m − M ( m ; d 1 ,d 2 ) − 1 , and there is at least one co dewo rd whose w eigh t is not divisib le by 2 m − M ( m ; d 1 ,d 2 ) . The follo win g prop osition w ill b e p ro ve n in Section 4. Prop osition 2: M ( m ; 3 , 13) = ( m − 1) / 2. By Prop ositions 1 and 2 and Lemmas 2 an d 5, w e obtain the follo win g theorem as the main result in this pap er. Theorem 1: F or an y o d d in teger m ≥ 5, the cod e C 1 , 3 , 13 has the same w eight distribution as th e binary triple-error-correcting p rimitiv e BCH co d e C 1 , 3 , 5 . 3 Minim um Distance of C 1 , 3 , 13 Pro of of Proposition 1: Let c = ( c 0 , c 1 , . . . , c n − 1 ) b e an arbitrary co dew ord in C 1 , 3 , 13 , where n = 2 m − 1. The Discrete F ourier T ransform of c is the sequence { A λ } w ith A λ = n − 1 X i =0 c i α iλ , 0 ≤ λ < n. 6 F rom the ab ov e formula, we ha v e that n is a p erio d of the sequen ce { A λ } . If A 5 = 0, then c is a co dew ord of the co de C 1 , 3 , 5 whic h h as minimum d istance 7 [20]. T his sho ws wt ( c ) ≥ 7. If A 9 = 0, then c is a co d ew ord of th e co de C 1 , 3 , 9 whic h also has minimum d istance 7 [20]. Consequently , wt ( c ) ≥ 7. Thus w e can assume that A 5 A 9 6 = 0 in th e follo wing analysis. By [26], the Hamming weig ht of c equals to the linear complexit y (also called linear span ) of the sequence { A λ } . It is sufficien t to pro v e that the rank of M is at least 7, w here M = A 0 A 1 · · · A n − 1 A 1 A 2 · · · A 0 . . . . . . . . . A n − 1 A 0 · · · A n − 2 . (4) T o this end, w e will argue separately acco rd ing to the parit y of w t ( c ). (1) Supp ose that wt ( c ) is od d, i.e., A 0 = 1. In this case, w e will fi nd t wo s u bmatrices M 1 and M 2 of M su ch that either M 1 or M 2 has full r an k , where M 1 = A 0 A 1 A 2 A 4 A 6 A 8 A 1 A 2 A 3 A 5 A 7 A 9 A 2 A 3 A 4 A 6 A 8 A 10 A 3 A 4 A 5 A 7 A 9 A 11 A 5 A 6 A 7 A 9 A 11 A 13 A 6 A 7 A 8 A 10 A 12 A 14 and M 2 = A 0 A 1 A 3 A 4 A 7 A 8 A 1 A 2 A 4 A 5 A 8 A 9 A 2 A 3 A 5 A 6 A 9 A 10 A 3 A 4 A 6 A 7 A 10 A 11 A 4 A 5 A 7 A 8 A 11 A 12 A 5 A 6 A 8 A 9 A 12 A 13 . Notice that A λ = 0 if λ ∈ C 1 ∪ C 3 ∪ C 13 , wh ere C i denotes the cycloto mic coset mo d u lo 2 m − 1 conta in in g the in teger i . Consequen tly , w e ha ve A 1 = A 2 = A 3 = A 4 = A 6 = A 8 = A 12 = A 13 = 0. F r om the expression of A λ , we ha ve A 10 = A 2 5 , A 14 = A 2 7 and A 18 = A 2 9 . It can b e d irectly verified that det( M 1 ) = A 2 5 A 7 ( A 3 7 + A 2 5 A 11 ) and det( M 2 ) = A 2 5 ( A 2 5 A 2 9 + A 5 A 9 A 2 7 + A 2 5 A 7 A 11 ) . If A 7 = 0, then det( M 2 ) = A 4 5 A 2 9 6 = 0 by our assumption that A 5 A 9 6 = 0, i.e., rank( M 2 ) = 6. If A 7 6 = 0 and A 11 = 0, then det ( M 1 ) 6 = 0 b y A 5 A 7 6 = 0, i.e., M 1 has rank 6. If A 7 6 = 0, A 11 6 = 0 and d et( M 1 ) = 0, then A 3 7 = A 2 5 A 11 . Thus, det( M 2 ) = A 2 5 ( A 2 5 A 2 9 + A 5 A 9 A 2 7 + A 4 7 ) , (5) whic h is either A 3 5 A 9 A 2 7 6 = 0 if A 5 A 9 = A 2 7 or A 2 5 ( A 5 A 9 + A 2 7 ) − 1 h ( A 5 A 9 ) 3 + ( A 2 7 ) 3 i 6 = 0 7 since gcd(3 , n ) = 1 if A 5 A 9 6 = A 2 7 . Th erefore, either M 1 or M 2 has full rank, and then rank( M ) ≥ 6. As a consequence, wt ( c ) ≥ 7. (2) Supp ose that wt ( c ) is ev en, i.e., A 0 = 0. If A 7 = 0, w e will p r o v e the follo wing submatrix M 3 = A 0 A 1 A 2 A 4 A 5 A 6 A 8 A 1 A 2 A 3 A 5 A 6 A 7 A 9 A 2 A 3 A 4 A 6 A 7 A 8 A 10 A 4 A 5 A 6 A 8 A 9 A 10 A 12 A 5 A 6 A 7 A 9 A 10 A 11 A 13 A 7 A 8 A 9 A 11 A 12 A 13 A 15 A 8 A 9 A 10 A 12 A 13 A 14 A 16 has rank 7. By a d irect calculation, we ha v e det( M 3 ) = A 7 5 A 2 9 6 = 0. T h us rank( M 3 ) ≥ 7 whic h implies that w t ( c ) ≥ 7. If A 7 6 = 0, w e will p r o v e the submatrix M 4 = A 0 A 1 A 2 A 4 A 6 A 7 A 8 A 1 A 2 A 3 A 5 A 7 A 8 A 9 A 2 A 3 A 4 A 6 A 8 A 9 A 10 A 4 A 5 A 6 A 8 A 10 A 11 A 12 A 5 A 6 A 7 A 9 A 11 A 12 A 13 A 8 A 9 A 10 A 12 A 14 A 15 A 16 A 12 A 13 A 14 A 16 A 18 A 19 A 20 has rank 7. By a direct calculation, w e ha ve det( M 4 ) = A 5 5 A 7 ( A 2 5 A 2 9 + A 5 A 9 A 2 7 + A 4 7 ). With a similar analysis as f or (5), w e ha ve det( M 4 ) 6 = 0 and then rank( M ) ≥ 7. Th us, w t ( c ) ≥ 7. Remark 1: The reference [26] sho we d that the minimum distance of a linear cyclic cod e is equal to the rank of a matrix constructed b y usin g Discrete F ourier T r ansform. This together with BCH or HT b ound established a lo wer b ound on the minimum d istance of the co de prop osed in [10]. In Prop osition 1, we apply th is metho d and the r esults for the m inim um d istances of the cyclic co des C 1 , 3 , 5 and C 1 , 3 , 9 [20] to obtain a lo w er b ound on minim um distance of C 1 , 3 , 13 . 4 Divisibilit y of W eigh ts in C ⊥ 1 , 3 , 13 In th is section, for an o dd integ er m = 2 k + 1 with k ≥ 2, we will pro ve M ( m ; 3 , 13) = k . 8 Let s , a and b b e in tegers with 0 ≤ s , a , b ≤ 2 m − 1, s ≡ 3 a + 13 b (mo d 2 m − 1), and assume that at least one of a and b is nonzero mo du lo 2 m − 1. Let s = m − 1 P i =0 s i 2 i , a = m − 1 P i =0 a i 2 i , and b = m − 1 P i =0 b i 2 i b e th e binary expressions of s , a and b , resp ectiv ely . W e first prov e M ( m ; 3 , 13) ≤ k , namely w ( s ) − w ( a ) − w ( b ) ≤ k in the sequel. Notice that 2 a , 8 b , 4 b (mo d 2 m − 1) h a v e the bin ary expressions m − 1 P i =0 a i − 1 2 i , m − 1 P i =0 b i − 3 2 i , m − 1 P i =0 b i − 2 2 i , r esp ectiv ely , and s ≡ 3 a + 13 b ≡ 2 a + a + 8 b + 4 b + b (mo d 2 m − 1). T aking d l = 1 for l ∈ { 1 , 2 , 3 , 4 , 5 } and a (1) = 2 a , a (2) = a , a (3) = 8 b , a (4) = 4 b , a (5) = b an d app lying Lemma 4, there are carries c i ∈ { 0 , 1 , 2 , 3 , 4 } suc h that 2 c i + s i = a i − 1 + a i + b i − 3 + b i − 2 + b i + c i − 1 , 0 ≤ i ≤ m − 1 , (6) where the subs cripts are take n mo du lo m . With w ( c ) = m − 1 P i =0 c i , by the m equalities in (6) w e ha ve w ( c ) + w ( s ) = 2 w ( a ) + 3 w ( b ) . (7) Let ν i = a i − 1 + a i + b i − 3 + b i − 2 + b i − 1 + b i − c i − 1 − c i , 0 ≤ i ≤ m − 1 (8) and w ( ν ) = m − 1 P i =0 ν i . Then by (8) and (7), w e ha v e w ( ν ) = 2 w ( a ) + 4 w ( b ) − 2 w ( c ) = 2 w ( s ) − w ( a ) − w ( b ) . (9) T o pro ve w ( s ) − w ( a ) − w ( b ) ≤ k , by (9) it is sufficien t to p r o v e w ( ν ) ≤ m . T o this end, w e will d efi ne a certain weigh ted directed graph D and r ecall some r elated defin itions in [5] as b elo w. A dir e cte d gr aph D is an ord er ed pair ( V ( D ) , A ( D )) consisting of a set V ( D ) of v ertices an d a set A ( D ), disjoin t fr om V ( D ), of ar cs , together with an incidenc e func tion ψ D that asso ciates with eac h arc ϑ of D an ordered pair of (not necessarily d istinct) vertic es ψ D ( ϑ ) = ( T ( ϑ ) , H ( ϑ )) of D . T he v ertex T ( ϑ ) is the tail of ϑ , and the v ertex H ( ϑ ) its he a d . F or eac h arc ϑ in a directed graph D , we can associate a r eal n umb er w ( ϑ ) with ϑ , and w ( ϑ ) is called its weight . In this case, D is called to b e a weighte d dir e cte d gr aph . In a directed graph D , a dir e cte d walk is an alternating sequence of v ertices and arcs W := P 0 ϑ 0 P 1 · · · P l − 1 ϑ l − 1 P l 9 suc h that for eac h i with 1 ≤ i ≤ l , P i − 1 and P i are the tail and head of ϑ i − 1 , resp ectiv ely . In this case, we refer to W as a dir e cte d ( P 0 , P l ) -walk . F or t wo ve rtices P i and P j in the walk W where 0 ≤ i < j ≤ l , the ( P i , P j )- se gment of W is the subsequ en ce of W starting with P i and ending with P j , an d it is denoted P i W P j . The d irected w alk W in D is close d if its initial and terminal vertices P 0 , P l are id en tical. With these pr ep arations, we can define a w eight ed directed graph D . The vertic es of D consist of all ve ctors P = ( x, y , z , u ), where x, y , z ∈ { 0 , 1 } and u ∈ { 0 , 1 , 2 , 3 , 4 } . Let P 1 = ( x 1 , y 1 , z 1 , u 1 ) and P 2 = ( x 2 , y 2 , z 2 , u 2 ) b e t wo v ertices of D , and define an arc ϑ w ith T ( ϑ ) = P 1 and H ( ϑ ) = P 2 if x 1 + y 1 + z 1 + x 2 + z 2 − 2 u 1 + u 2 = 0 , or 1 . (10) The weigh t of the arc ϑ is defined as w ( ϑ ) = x 1 + y 1 + z 1 + x 2 + y 2 + z 2 − u 1 − u 2 . Th u s for i ∈ { 0 , 1 , · · · , m − 1 } , V i = ( a i , b i , b i − 2 , c i ) (11) are m vertice s of D , w here a i , b i , and c i are those integ ers in (6). F urthermore, there are m arcs ϑ i with w ( ϑ i ) = ν i defined b y (8) with the tail V i = ( a i , b i , b i − 2 , c i ) and head V i − 1 = ( a i − 1 , b i − 1 , b i − 3 , c i − 1 ) for all 0 ≤ i ≤ m − 1 s in ce a i + b i + b i − 2 + a i − 1 + b i − 3 − 2 c i + c i − 1 = s i ∈ { 0 , 1 } b y (6), w here the subscripts are tak en mo du lo m . With th e help of a computer, w e h a v e that th ere are totall y 320 arcs in D , and their we ight distribution is giv en in T able 4. F urthermore, every ve rtex in the set Γ = n (1 , 1 , 0 , 0) , (1 , 0 , 1 , 0) , (0 , 1 , 1 , 0) , (1 , 1 , 1 , 0 ) o (12) cannot b e the tail of any arc in D . Some arcs ϑ with head H ( ϑ ) 6∈ Γ will b e used in this section and th ey are listed in App endix A. T able 4: The wei ght d istribution of all arcs in the wei ghted d irected graph D W eigh t -6 -5 -4 -3 -2 -1 0 1 2 3 4 The num b er of arcs 1 16 36 43 43 42 43 43 36 16 1 Notice that f or the case ν i < 2 for all i ∈ { 0 , 1 , · · · , m − 1 } , it can b e easily verified that w ( ν ) ≤ m . Cons equ en tly , the pr o of for w ( ν ) ≤ m can b e pro ceeded in t wo steps as b elo w. Step 1 : T o pr o v e that for an y ν i ≥ 2, there exists a p ositiv e int eger t ≤ m suc h that ν i + ν i − 1 + · · · + ν i − t +1 ≤ t . 10 Step 2 : Based on Step 1, we will p r o v e w ( ν ) = m − 1 P i =0 ν i ≤ m . The tw o steps are s ummarized as the f ollo win g Prop ositions 3 and 4. Prop osition 3: F or any ν i ≥ 2, there exists a p ositiv e in teger t ≤ m suc h that ν i + ν i − 1 + · · · + ν i − t +1 ≤ t , where th e sub s cripts are tak en mo d u lo m . By the wei ghted directed graph D defined as ab o v e, the n u m b er ν i + ν i − 1 + · · · + ν i − t +1 can b e regarded as the sum of the we ights of some arcs in D . T o fin ish the pro of of P rop osition 3, w e need to study a set P = n W = P 0 ϑ 0 P 1 ϑ 1 · · · P i − 1 ϑ i − 1 P i ϑ i · · · P q − 1 ϑ q − 1 P q | W ∈ D , q is dep endent on W o (13) consisting of all directed walks W with th e follo w ing prop erties: (I) an y verte x of the set Γ in (12) do es not occur in W ; (I I) f or 0 ≤ i ≤ q − 2, any three consecutiv e ve r tices P i , P i +1 , and P i +2 in W satisfy P i (3) = P i +2 (2), wh er e P i ( l ) denotes the l -th comp onent of P i for l ∈ { 1 , 2 , 3 , 4 } ; in addition, if the w alk W is closed, then P q − 1 (3) = P 1 (2); (I I I) any arc ϑ i in W satisfies that w ( ϑ i ) ≥ ( i + 2) − T i for 0 ≤ i ≤ q − 1, where T 0 = 0 and T i = i − 1 P l =0 w ( ϑ l ) for i ≥ 1. If Pr op osition 3 cannot b e true, then there is an intege r i 0 with 0 ≤ i 0 ≤ m − 1 suc h that ν i 0 ≥ 2 and ν i 0 + ν i 0 − 1 + · · · + ν i 0 − t +1 ≥ t + 1 for an y p ositiv e in teger t w ith 2 ≤ t ≤ m . Let W 0 = P 0 ϑ 0 P 1 ϑ 1 · · · P i − 1 ϑ i − 1 P i ϑ i · · · P m − 2 ϑ m − 2 P m − 1 (14) b e th e w alk su c h that P i = V i 0 − i in (11) f or 0 ≤ i ≤ m − 1, and ϑ i b e th e arc with T ( ϑ i ) = P i and H ( ϑ i ) = P i +1 for i ∈ { 0 , 1 , · · · , m − 1 } , where the subscripts are tak en mo dulo m . Then, w e h a v e w ( ϑ 0 ) ≥ 2 and for any p ositiv e in teger t w ith 2 ≤ t ≤ m s uc h that w ( ϑ 0 ) + w ( ϑ 1 ) + · · · + w ( ϑ t − 1 ) ≥ t + 1. Thus by (11) and the analysis th er ein, W 0 ∈ P and it is closed. As a consequence, it will lead to a con tradiction if any w alk W ∈ P is not closed. In fact, w e can prov e that an y walk W ∈ P is not closed in the sequel. This will giv e the pro of of Prop osition 3. The follo win g notations are used throughout this section: • P i ( η, ω ) − − − → denotes any walk P i ϑ i P i +1 with T ( ϑ i ) = P i , H ( ϑ i ) = P i +1 , P i +1 (2) = η and w ( ϑ i ) ≥ ω ; • P i ( − , ω ) − − − → denotes any wa lk P i ϑ i P i +1 with T ( ϑ i ) = P i , H ( ϑ i ) = P i +1 , P i +1 (2) ∈ { 0 , 1 } and w ( ϑ i ) ≥ ω ; 11 • P i ( η, ω ) − − − → O denotes that there do es not exist any arc ϑ su c h that T ( ϑ ) = P i , H ( ϑ ) ∈ D , ( H ( ϑ ))(2) = η and w ( ϑ ) ≥ ω . With th e ab o v e notations, w e can con ve n ien tly d escrib e the w alks in P . Example 1: Let q b e a p ositiv e integ er and ω = ( j + 2) − T j = 1 for some p ositiv e inte ger j with 0 ≤ j < q , and let W : P 0 − → P 1 − → · · · − → P j − 1 − → P j = (0 , 0 , 0 , 0) (0 , ω ) − − − → P j +1 − → P j +2 − → · · · − → P q b e a w alk in the set P , and ϑ i b e the arc with the ta il P i and h ead P i +1 for eac h i ∈ { 0 , 1 , · · · , q − 1 } . By App endix A, w e can fin d all p ossibilities for the segmen t P j +1 W P q , which is completely determined by the w alk (0 , 0 , 0 , 0) (0 , 1) − − − → . If we find all p ossibilities for the segmen t P j +1 W P q , then w e also know all p ossibilities for the segmen t P j +1 W P q ′ for an y in teger j + 1 ≤ q ′ ≤ q . Therefore, without loss of generalit y , we can assu me that the inte ger q is large enough. Since P j +1 (2) = 0 and w ( ϑ j ) ≥ 1, by App endix A, w e ha v e P j +1 ∈ (1 , 0 , 0 , 0) , (0 , 0 , 1 , 0) . If P j +1 = (1 , 0 , 0 , 0), b y Prop erties (I I) and (I I I) of the walks in P , we ha v e P j +2 (2) = P j (3) = 0 and w ( ϑ j +1 ) ≥ ( j + 3) − T j +1 = ( j + 3) − w ( ϑ j ) − T j = ( j + 2) − w ( ϑ j ) − T j = 1. By App end ix A, we can uniqu ely determine P j +2 = (0 , 0 , 0 , 0). F urthermore, with w ( ϑ j +1 ) = 1 and P j +1 = (1 , 0 , 0 , 0), w e ha v e w ( ϑ j +2 ) ≥ ( j + 4) − T j +2 = ( j + 4) − w ( ϑ j +1 ) − T j +1 = ( j + 3) − T j +1 = 1 (15) and P j +3 (2) = 0. Therefore, for P j +1 = (1 , 0 , 0 , 0), P j W P j +3 can b e expressed as (0 , 0 , 0 , 0) (0 , 1) − − − → (1 , 0 , 0 , 0) (0 , 1) − − − → (0 , 0 , 0 , 0) (0 , 1) − − − → . (16) Similarly , for P j +1 = (0 , 0 , 1 , 0), P j W P j +5 is giv en by (0 , 0 , 0 , 0) (0 , 1) − − − → (0 , 0 , 1 , 0) (0 , 1) − − − → (0 , 0 , 0 , 0) (1 , 1) − − − → (0 , 1 , 0 , 0) (0 , 1) − − − → (0 , 0 , 0 , 0) (0 , 1) − − − → . (17) Com binin g (16) and (17), we ha v e an expression consisting of t wo segmen ts with initial v ertex P j (0 , 0 , 0 , 0) (0 , 1) − − − → (1 , 0 , 0 , 0) (0 , 1) − − − → (0 , 0 , 0 , 0) (0 , 1) − − − → (0 , 0 , 1 , 0) (0 , 1) − − − → (0 , 0 , 0 , 0) (1 , 1) − − − → (0 , 1 , 0 , 0) (0 , 1) − − − → (0 , 0 , 0 , 0) (0 , 1) − − − → . (18) In the fi r st segmen t of (18), P j +3 = (1 , 0 , 0 , 0) or (0 , 0 , 1 , 0) since (0 , 0 , 0 , 0) (0 , 1) − − − → has only t wo p ossible forms, wh ic h hav e o ccurred as P j W P j +1 in the first and second segmen ts of (18), 12 resp ectiv ely . By a similar analysis, w e ha v e P j +5 = (1 , 0 , 0 , 0) or (0 , 0 , 1 , 0) in the second segment of (18). T herefore, again b y (18), we hav e that P j +3 W P j +5 has the form as (1 , 0 , 0 , 0) (0 , 1) − − − → (0 , 0 , 0 , 0) (0 , 1) − − − → (19) or P j +3 W P j +7 has the form as (0 , 0 , 1 , 0) (0 , 1) − − − → (0 , 0 , 0 , 0) (1 , 1) − − − → (0 , 1 , 0 , 0) (0 , 1) − − − → (0 , 0 , 0 , 0) (0 , 1) − − − → (20) in the fi rst segment of (18). Similarly , we ha v e that P j +5 W P j +7 has the form as (19) or P j +5 W P j +9 has the form as (20) in the second segmen t of (18). Rep eating th e ab ov e p r o cess, all p ossibilities of P j +1 W P q can b e obtained. F ur ther, all vertice s P l ( j ≤ l ≤ q ) ha ve o ccurr ed in the t wo segmen ts of (18), and they are (0 , 0 , 0 , 0), (1 , 0 , 0 , 0), (0 , 1 , 0 , 0) , and (0 , 0 , 1 , 0) . Remark 2: In Example 1, (0 , 0 , 0 , 0) (0 , 1) − − − → completely determines all p ossibilities for the segmen t P j +1 W P q of W . Th e expression (18) consists of tw o basic segments of W , by whic h all p ossibilities of the segmen t P j +1 W P q can b e con v eniently found . In the pro ofs of Lemmas 6 an d 7, for some giv en P j ( η,ω ) − − − → of a w alk W in P , w e will frequent ly need to determine all p ossibilities for the segmen t P j +1 W P q of W . Similarly as in Example 1, we w ill use s ome expression consisting of b asic segments of W to determine all p ossibilities of P j +1 W P q . W e call the expression as (18 ) a set of b asic se gments (SBS) of P j ( η,ω ) − − − → . The follo win g t w o lemmas will b e used to prov e Prop osition 3. Lemma 6: Let q b e a p ositiv e integer and ω = ( j + 2) − T j for some p ositiv e inte ger j with 0 ≤ j < q . F or any wal k W : P 0 − → P 1 − → · · · − → P j − 1 − → P j = (0 , 0 , 0 , 0) ( − , ω ) − − − → P j +1 − → · · · − → P q − 1 − → P q in the set P defined by (13), we hav e (i) if ω = 0 or 1, all ve r tices P l ( j + 1 ≤ l ≤ q ) o ccurr ing in th e w alk W are conta ined in th e set S 1 = n (0 , 0 , 0 , 0) , (0 , 0 , 1 , 0) , (0 , 1 , 0 , 0) , (1 , 0 , 0 , 0) , (0 , 1 , 0 , 1) o ; (21) (ii) if ω = − 1, all v ertices P l ( j + 1 ≤ l ≤ q ) o ccurring in the walk W are cont ained in the set S 2 = S 1 S n (0 , 0 , 0 , 1) , (0 , 0 , 1 , 1) , (1 , 0 , 0 , 1) o ; (22) (iii) if ω = − 2, all v ertices P l ( j + 1 ≤ l ≤ q ) o ccurr in g in the walk W are contai n ed in the set S 3 = S 2 S n (1 , 0 , 1 , 1) , (0 , 1 , 1 , 1) , (1 , 1 , 0 , 1) o . (23) 13 The p ro of of Lemma 6 is p r esen ted in App endix B. Lemma 7: F or the walk W : P 0 − → P 1 − → · · · − → P q − 1 − → P q in the set P , if the initial v ertex P 0 ∈ (1 , 0 , 0 , 0) , (0 , 1 , 0 , 0) , (0 , 0 , 1 , 0) , (1 , 0 , 1 , 1 ) , (1 , 1 , 0 , 1), (0 , 1 , 1 , 1) , then W cannot b e closed. Pro of: Let ϑ j denote th e arc with the tail P j and h ead P j +1 for eac h j ∈ { 0 , 1 , · · · , q − 1 } . Since W ∈ P , b y Prop ert y (I I I) of th e w alks in P , we ha v e w ( ϑ 0 ) ≥ 2. If W is closed, then we m ust ha ve P q = P 0 and P q − 1 (3) = P 1 (2). T h e lemma is pr ov en according to six cases of the v ertex P 0 as follo ws. If P 0 = (1 , 0 , 0 , 0) and w ( ϑ 0 ) ≥ 2, th en P 1 = (0 , 1 , 0 , 0) by App endix A. Con s equen tly , P 2 (2) = 0 and by Prop er ty (I I I) of the wa lks in P , w ( ϑ 1 ) ≥ 1. By a similar analysis as in Example 1, (0 , 1 , 0 , 0) (0 , 1) − − − → has an SBS as (0 , 1 , 0 , 0) (0 , 1) − − − → (0 , 0 , 0 , 0) (0 , 1) − − − → (1 , 0 , 0 , 0) (0 , 1) − − − → (0 , 0 , 0 , 0) (0 , 1) − − − → (0 , 0 , 1 , 0) (0 , 1) − − − → (0 , 0 , 0 , 0) (1 , 1) − − − → (0 , 1 , 0 , 0) (0 , 1) − − − → . (24) F rom (24), we can kn ow that all vertic es and arcs in P 1 W P q ha ve o ccurr ed in (24). If P q = P 0 = (1 , 0 , 0 , 0), then b y (24), P q − 1 = (0 , 0 , 0 , 0) and then P q − 1 (3) = 0 6 = P 1 (2). Therefore the w alk W cannot b e closed if P 0 = (1 , 0 , 0 , 0). The case P 0 = (0 , 1 , 0 , 0) can b e similarly pro ven as the case P 0 = (1 , 0 , 0 , 0). If P 0 = (0 , 0 , 1 , 0), then P 0 W P 4 has the form as (0 , 0 , 1 , 0) ( − , 2) − − − → (0 , 1 , 0 , 0) (1 , 1) − − − → (0 , 1 , 0 , 0) (0 , 0) − − − → (0 , 0 , 0 , 0) (0 , 0) − − − → . (25) If W is closed, then P q = (0 , 0 , 1 , 0) and P q − 1 (3) = 1. By (25), w e ha ve q ≥ 5. By Lemma 6 (i), the vertic es P j for 4 ≤ j ≤ q in W are con tained in S 1 . C on s equen tly , P q − 1 ∈ S 1 . Notice that (0 , 0 , 1 , 0) is the uniqu e ve r tex w ith the th ir d comp onent 1 in the set S 1 . As a consequence, P q − 1 = (0 , 0 , 1 , 0) and the arc ϑ q − 1 is (0 , 0 , 1 , 0) − → (0 , 0 , 1 , 0), wh ic h do es not exist by App endix A. T his leads to a con tradiction and th en W cannot b e closed. If P 0 = (1 , 0 , 1 , 1), then (1 , 0 , 1 , 1) ( − , 2) − − − → has an S BS as (1 , 0 , 1 , 1) ( − , 2) − − − → (1 , 0 , 0 , 0) (1 , 1) − − − → (0 , 1 , 0 , 0) (0 , 0) − − − → (0 , 0 , 0 , 0) (0 , 0) − − − → (0 , 1 , 0 , 0) (1 , 1) − − − → (0 , 1 , 0 , 0) (0 , 0) − − − → (0 , 0 , 0 , 0) (0 , 0) − − − → (0 , 0 , 1 , 0) (1 , 1) − − − → (0 , 1 , 0 , 0) (1 , 0) − − − → (0 , 1 , 0 , 0) (0 , − 1) − − − − → (0 , 0 , 0 , 0) (0 , − 1) − − − − → . (26) 14 The v ertices P j for 4 ≤ j ≤ q of the first and second segment s of (26) are con tained in S 1 and the ve r tices P j for 5 ≤ j ≤ q of the third segment in (26 ) are con tained in S 2 b y Lemma 6 (i) and (ii). Notice that (1 , 0 , 1 , 1) 6∈ S 1 and (1 , 0 , 1 , 1) 6∈ S 2 . Consequentl y , the walk W cannot b e closed. If P 0 = (1 , 1 , 0 , 1), then P 0 W P 3 has thr ee p ossible f orm s as (1 , 1 , 0 , 1) ( − , 2) − − − → (1 , 0 , 0 , 0) (0 , 1) − − − → (0 , 0 , 0 , 0) (0 , 1) − − − → (0 , 1 , 0 , 0) (0 , 1) − − − → (0 , 0 , 0 , 0) (0 , 1) − − − → (0 , 0 , 1 , 0) (0 , 1) − − − → (0 , 0 , 0 , 0) (1 , 1) − − − → . The v ertices P j for 3 ≤ j ≤ q are con tained in S 1 b y Lemma 6 (i). The fact (1 , 1 , 0 , 1) 6∈ S 1 implies that W cannot b e closed. The case P 0 = (0 , 1 , 1 , 1) can b e similarly pro ven as the case P 0 = (1 , 0 , 1 , 1). The p ro of is finished. Applying L emm as 6 and 7, w e will finish the p ro of of Prop osition 3 as b elo w. Pro of of Proposition 3: I f the result is not true, the walk W 0 defined in (14) b elongs to the set P and w ( ϑ 0 ) ≥ 2. W e will pro v e that W 0 cannot b e closed acco r ding to ϑ 0 . Notice that there are no arcs ϑ with tail T ( ϑ ) ∈ Γ, where Γ is defined by (12). A s a consequence, W 0 cannot b e closed if ϑ 0 o ccurs in T able 5. T able 5: All arcs ϑ with w ( ϑ ) ≥ 2 and H ( ϑ ) ∈ Γ T ( ϑ ) H ( ϑ ) w ( ϑ ) T ( ϑ ) H ( ϑ ) w ( ϑ ) T ( ϑ ) H ( ϑ ) w ( ϑ ) (0,0,0, 0) (1,1,0,0) 2 (0,0,0 ,0) (0,1,1,0 ) 2 (0,0 ,0,1) (1,1,1 ,0) 2 (1,0,0, 1) (1,1,0,0) 2 (1,0,0 ,1) (1,0,1,0 ) 2 (1,0 ,0,1) (0,1,1 ,0) 2 (1,0,0, 1) (1,1,1,0) 3 (0,1,0 ,1) (1,1,0,0 ) 2 (0,1 ,0,1) (1,0,1 ,0) 2 (0,1,0, 1) (0,1,1,0) 2 (0,1,0 ,1) (1,1,1,0 ) 3 (1,1 ,0,1) (1,1,0 ,0) 3 (1,1,0, 1) (0,1,1,0) 3 (1,1,0 ,2) (1,0,1,0 ) 2 (1,1 ,0,2) (1,1,1 ,0) 3 (0,0,1, 1) (1,1,0,0) 2 (0,0,1 ,1) (1,0,1,0 ) 2 (0,0 ,1,1) (0,1,1 ,0) 2 (0,0,1, 1) (1,1,1,0) 3 (1,0,1 ,1) (1,1,0,0 ) 3 (1,0 ,1,1) (0,1,1 ,0) 3 (1,0,1, 2) (1,0,1,0) 2 (1,0,1 ,2) (1,1,1,0 ) 3 (0,1 ,1,1) (1,1,0 ,0) 3 (0,1,1, 1) (0,1,1,0) 3 (0,1,1 ,2) (1,0,1,0 ) 2 (0,1 ,1,2) (1,1,1 ,0) 3 (1,1,1, 2) (1,1,0,0) 3 (1,1,1 ,2) (1,0,1,0 ) 3 (1,1 ,1,2) (0,1,1 ,0) 3 (1,1,1, 2) (1,1,1,0) 4 W e list all arcs ϑ with w ( ϑ ) ≥ 2, T ( ϑ ) 6∈ S 3 and H ( ϑ ) 6∈ Γ in T able 6, wh ere S 3 is defin ed 15 b y (23). If ϑ 0 is the arc (1 , 1 , 0 , 2) − → (1 , 1 , 1 , 1) in T able 6, by App end ix A, P 0 W 0 P 3 has the form as (1 , 1 , 0 , 2) − → (1 , 1 , 1 , 1) (0 , 1) − − − → (0 , 0 , 0 , 0) (1 , 0) − − − → . The ve r tices P j for j ≥ 3 are con tained in S 1 b y Lemma 6 (i). Notice that (1 , 1 , 0 , 2) 6∈ S 1 . Consequent ly , W 0 cannot b e closed. T able 6: All arcs ϑ with w ( ϑ ) ≥ 2, T ( ϑ ) 6∈ S 3 and H ( ϑ ) 6∈ Γ T ( ϑ ) H ( ϑ ) w ( ϑ ) T ( ϑ ) H ( ϑ ) w ( ϑ ) T ( ϑ ) H ( ϑ ) w ( ϑ ) (1,1,0, 2) (1,1,1,1) 2 (1,0,1 ,2) (1,1,1,1 ) 2 (0,1 ,1,2) (1,1,1 ,1) 2 (1,1,1, 3) (1,1,1,1) 2 (1,1,1 ,1) (0,0,0,0 ) 2 (1,1 ,1,1) (0,1,0 ,0) 3 (1,1,1, 2) (1,1,0,1) 2 (1,1,1 ,2) (0,1,1,1 ) 2 (1,1 ,1,2) (1,0,0 ,0) 2 (1,1,1, 2) (0,0,1,0) 2 If ϑ 0 is the arc (1 , 0 , 1 , 2) − → (1 , 1 , 1 , 1) in T able 6, P 0 W 0 P 5 has the form as (1 , 0 , 1 , 2) − → (1 , 1 , 1 , 1) (1 , 1) − − − → (0 , 1 , 0 , 0) (1 , − 1) − − − − → (0 , 1 , 0 , 0) (0 , − 2) − − − − → (0 , 0 , 0 , 0) (0 , − 2) − − − − → . The vertices P j for j ≥ 5 are cont ained in S 3 b y Lemm a 6 (iii). Therefore, W 0 cannot b e closed since (1 , 0 , 1 , 2) 6∈ S 3 . Th e cases f or the arcs (0 , 1 , 1 , 2) − → (1 , 1 , 1 , 1) and (1 , 1 , 1 , 3) − → (1 , 1 , 1 , 1) in T able 6 can b e similarly pro ven. If ϑ 0 is the arc (1 , 1 , 1 , 1) − → (0 , 0 , 0 , 0) in T able 6, then P 0 W 0 P 2 has the form as (1 , 1 , 1 , 1) − → (0 , 0 , 0 , 0) (1 , 1) − − − → . Th u s, all ve rtices P j for j ≥ 2 are con tained in S 1 b y Lemma 6 (i), and then W 0 cannot b e closed since (1 , 1 , 1 , 1) 6∈ S 1 . If ϑ 0 is the arc (1 , 1 , 1 , 1) − → (0 , 1 , 0 , 0) in T able 6, P 0 W 0 P 4 has the form as (1 , 1 , 1 , 1) − → (0 , 1 , 0 , 0) (1 , 0) − − − → (0 , 1 , 0 , 0) (0 , − 1) − − − − → (0 , 0 , 0 , 0) (0 , − 1) − − − − → , and the vertic es P j for j ≥ 4 are cont ained in S 2 b y L emma 6 (ii). So W 0 cannot b e closed since (1 , 1 , 1 , 1) 6∈ S 2 . If ϑ 0 is the arc (1 , 1 , 1 , 2) − → (1 , 1 , 0 , 1) in T able 6, (1 , 1 , 0 , 1) (1 , 1) − − − → has an SBS as (1 , 1 , 0 , 1) (1 , 1) − − − → (0 , 1 , 0 , 0) (0 , 0) − − − → (0 , 0 , 0 , 0) (0 , 0) − − − → (0 , 1 , 0 , 1) (0 , 1) − − − → (1 , 0 , 0 , 0) (0 , 1) − − − → (0 , 0 , 0 , 0) (0 , 1) − − − → (0 , 0 , 1 , 0) (0 , 1) − − − → (0 , 0 , 0 , 0) (1 , 1) − − − → 16 and then the v ertices P j for j ≥ 4 are con tained in S 1 b y Lemma 6 (i). Thus W 0 cannot b e closed sin ce (1 , 1 , 1 , 2) 6∈ S 1 . If ϑ 0 is the arc (1 , 1 , 1 , 2) − → (0 , 1 , 1 , 1) in T able 6, (0 , 1 , 1 , 1) (1 , 1) − − − → has an SBS as (0 , 1 , 1 , 1) (1 , 1) − − − → (0 , 1 , 0 , 0) (1 , 0) − − − → (0 , 1 , 0 , 0) (0 , − 1) − − − − → (0 , 0 , 0 , 0) (0 , − 1) − − − − → (0 , 1 , 0 , 1) (1 , 1) − − − → (1 , 1 , 0 , 1) (0 , 1) − − − → (0 , 1 , 1 , 1) (0 , 1) − − − → . The w alks (0 , 1 , 1 , 1) (0 , 1) − − − → and (1 , 1 , 0 , 1) (0 , 1) − − − → ha ve b een analyzed in (35) and (36) in App endix B, resp ectiv ely . Th us by Lemma 6, the vertices P j for j ≥ 1 are con tained in S 3 . So W 0 cannot b e closed since (1 , 1 , 1 , 2) 6∈ S 3 . If ϑ 0 is the arc (1 , 1 , 1 , 2) − → (1 , 0 , 0 , 0) in T able 6, P 0 W 0 P 4 has the form as (1 , 1 , 1 , 2) − → (1 , 0 , 0 , 0) (1 , 1) − − − → (0 , 1 , 0 , 0) (0 , 0) − − − → (0 , 0 , 0 , 0) (0 , 0) − − − → , and th e ve r tices P 4 for j ≥ 4 are conta in ed in S 1 b y Lemma 6 (i). So W 0 cannot b e closed since (1 , 1 , 1 , 2) 6∈ S 1 . If ϑ 0 is the arc (1 , 1 , 1 , 2) − → (0 , 0 , 1 , 0) in T able 6, P 0 W 0 P 5 has the form as (1 , 1 , 1 , 2) − → (0 , 0 , 1 , 0) (1 , 1) − − − → (0 , 1 , 0 , 0) (1 , 0) − − − → (0 , 1 , 0 , 0) (0 , − 1) − − − − → (0 , 0 , 0 , 0) (0 , − 1) − − − − → . Th u s the v ertices P j for j ≥ 5 are cont ained in S 2 b y Lemma 6 (ii). So W 0 cannot b e closed since (1 , 1 , 1 , 2) 6∈ S 2 . The ab ov e facts sho w th at if ϑ 0 is an y arc in T able 6 th en the walk W 0 cannot b e closed. Supp ose th at ϑ 0 satisfies T ( ϑ 0 ) ∈ S 3 and H ( ϑ 0 ) 6∈ Γ, i.e., those arcs in T able 7. By Lemma 7, w e still h a v e that the walk W 0 cannot b e closed for an y ϑ 0 giv en by T able 7. Ho w ever, by (11) and the analysis therein, w e h a v e that W 0 is closed. This con tradiction sho ws that the assumption at the b eginnin g of the p ro of d o es not hold, and then th e pro of is fin ished. T able 7: All arcs ϑ with w ( ϑ ) ≥ 2, T ( ϑ ) ∈ S 3 and H ( ϑ ) 6∈ Γ T ( ϑ ) H ( ϑ ) w ( ϑ ) T ( ϑ ) H ( ϑ ) w ( ϑ ) T ( ϑ ) H ( ϑ ) w ( ϑ ) (1,0,0, 0) (0,1,0,0) 2 (0,1,0 ,0) (0,1,0,0 ) 2 (0,0 ,1,0) (0,1,0 ,0) 2 (1,0,1, 1) (1,0,0,0) 2 (1,0,1 ,1) (0,1,0,0 ) 2 (1,0 ,1,1) (0,0,1 ,0) 2 (1,1,0, 1) (1,0,0,0) 2 (1,1,0 ,1) (0,1,0,0 ) 2 (1,1 ,0,1) (0,0,1 ,0) 2 (0,1,1, 1) (1,0,0,0) 2 (0,1,1 ,1) (0,1,0,0 ) 2 (0,1 ,1,1) (0,0,1 ,0) 2 17 Remark 3: In the pro of of Pr op osition 3, we do not distinguish whether the v ertices of the w alk W 0 are in th e set { V 0 , V 1 , · · · , V m − 1 } or not. That is to sa y , w e ha ve prov en that eac h walk in P cann ot b e closed. Prop osition 4: F or the in teger sequence ν 0 , ν 1 , . . . , ν m − 1 of p erio d m , if for an y ν i ≥ 2, there exists a p ositiv e in teger t ≤ m suc h that ν i + ν i − 1 + · · · + ν i − t +1 ≤ t , then m − 1 P i =0 ν i ≤ m . Pro of: Let I = i | ν i ≥ 2 and | I | = p . Th u s, all elemen ts of I can b e listed as i 1 , i 2 , · · · , i p , where i 1 < i 2 < · · · < i p . F or eac h in teger i j ∈ I , ther e exists a least p ositiv e integ er t j suc h that ν i j + ν i j − 1 + · · · + ν i j − t j +1 ≤ t j , (27) and let N j = i j , i j − 1 , · · · , i j − t j + 1 b e a subset of Z m . T hen the in equalit y (27) can b e written as P i ∈ N j ν i ≤ t j = | N j | . Let N = p S j =1 N j , and w e ha ve that ν i ≤ 1 if i ∈ Z m \ N . If p = 1, ν i 1 + ν i 1 − 1 + · · · + ν i 1 − t 1 +1 ≤ t 1 . In this case, the pro of follo ws the fact that other ν j satisfies ν j ≤ 1. If p ≥ 2, we claim that for tw o intege rs j and j ′ with 1 ≤ j < j ′ ≤ p , th e sets N j and N j ′ are disjoint or one con taining another one. Without loss of generalit y , we tak e j = 1 and j ′ = 2. Then we ha ve ν i 1 + ν i 1 − 1 + · · · + ν i 1 − t 1 +1 ≤ t 1 and ν i 2 + ν i 2 − 1 + · · · + ν i 2 − t 2 +1 ≤ t 2 , (28) resp ectiv ely , where the subscrip ts are tak en mo dulo m since the intege r sequence has p er io d m . If the ab o v e claim is n ot true, then w e ha ve i 1 − t 1 + 1 < i 2 − t 2 + 1 ≤ i 1 < i 2 and consider the follo wing sequence ν i 1 − t 1 +1 , · · · , ν i 2 − t 2 +1 , ν i 2 − t 2 +2 , · · · , ν i 1 , · · · , ν i 2 . Notice that t 1 and t 2 are th e least p ositiv e in tegers satisfying (28). C onsequen tly , we ha ve ν i 2 − t 2 +1 + ν i 2 − t 2 +2 + · · · + ν i 1 > i 1 − i 2 + t 2 , and ν i 1 +1 + ν i 1 +2 + · · · + ν i 2 > i 2 − i 1 . This implies ν i 2 − t 2 +1 + ν i 2 − t 2 +2 + · · · + ν i 1 + ν i 1 +1 + ν i 1 +2 + · · · + ν i 2 > t 2 , whic h con tradicts with (28) and then th e claim is true. Th us there exists a subset J of th e set { 1 , 2 , · · · , p } su c h that N = [ j ∈ J N j and N j \ N j ′ = ∅ for an y tw o different elemen ts j and j ′ of J. 18 Th u s | N | = P j ∈ J | N j | = P j ∈ J t j and we ha ve that X i ∈ N ν i = X j ∈ J X i ∈ N j ν i ≤ X j ∈ J t j = | N | . Therefore, we ha ve m − 1 X i =0 ν i = X i ∈ Z m \ N ν i + X i ∈ N ν i ≤ X i ∈ Z m \ N 1 + | N | = m, and th is finishes th e pro of. Prop ositions 3 and 4 tell us that M ( m ; 3 , 13) ≤ k . F urthermore, w e can also pr o v e that the equal sign holds. Lemma 8: (Theorem 14, [17]) W e ha v e that M ( m ; 2 r + 1) = ( m/ 2 , if m/ ( r , m ) is ev en , ( m − ( m, r )) / 2 , if m/ ( r , m ) is o dd . Pro of of Prop osition 2: By P r op ositions 3 and 4, w e ha v e w ( ν ) ≤ m and then b y (9 ) M ( m ; 3 , 13) = max ( w ( s ) − w ( a ) − w ( b )) ≤ k where th e maxim um is ov er all inte gers s , a , b such that 0 ≤ s, a, b ≤ 2 m − 1 , s ≡ 3 a + 13 b (mod 2 m − 1) , a or b 6≡ 0 (mo d 2 m − 1) . On the other hand, w e ha ve M ( m ; 3 , 13) ≥ M ( m ; 3) b y the d efi nition of M ( m ; 3 , 13). Applying Lemma 8, we hav e k = ( m − ( m, r )) / 2 = M ( m ; 3) ≤ M ( m ; 3 , 13) ≤ k. Therefore, we ha ve M ( m ; 3 , 13) = k and the pro of is fin ished. 5 Concluding Remarks F or o dd m ≥ 5, a new triple-error-correcting cyclic co de of length 2 m − 1 has b een foun d. It is defined by zeros α, α 3 and α 13 , an d the exp onents 3 and 13 come from the Gold and Kasami- W elc h APN p o w er f unctions, resp ectiv ely . T o generalize the constru ction of the code C 1 , 3 , 13 , one can consid er the class of cyclic co des C w ith the dual co d es C ⊥ ha ving the form C ⊥ = n c ( ǫ, γ , δ ) = (T r m 1 ( ǫx + γ f ( x ) + δ g ( x )) x ∈ F ∗ 2 m | ǫ, γ , δ ∈ F 2 m o where f ( x ) and g ( x ) are d ifferen t APN functions from F 2 m to itself. If the p olynomial T r m 1 ( ǫx + γ f ( x ) + δ g ( x )) in v ariable x has algebraic degree greater than 2, some to ols other than the theory of quadratic forms are p ossibly n eeded. 19 References [1] E. Berlek amp, Algebraic Co ding T heory , New Y ork: McGra w-Hill, 1968. [2] E. Berlek amp, The we ight enumerators for certain su b co des of the seco n d order binary Reed- Muller codes, Inf. C ontr., vo l. 17, no. 5, pp. 485- 500, 1970. [3] E. Berle k amp, W eigh t en u meration theorems, in Pro c. Sixth Allerton Conf. Circuit an d Systems T heory , Urbana, IL, pp. 161-170, 1968. [4] T. Beth and C. Ding, On almost p erfect non lin ear p erm utations, in Ad v ances in Cryptography-EUR OC R YPT’93, Lecture Notes in Computer Science 765, Berlin, German y: Springer-V erlag, pp. 65-76, 1994. [5] J. Bondy and U. Murt y , Graph Theory , Berlin, Germany: S pringer-V erlag, 2008. [6] R. Bose and D. Ray- C h audhuri, On a class of error correcting binary group co des, In f. 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T ec hnol., Zur ic h, S witzerland, 1988. 21 App endix A: Some Arcs ϑ in D App end ix A giv es all arcs ϑ with th e tail T ( ϑ ) in the set n (0 , 0 , 0 , 0) , (0 , 0 , 0 , 1) , (1 , 0 , 0 , 0) , (1 , 0 , 0 , 1) , (0 , 1 , 0 , 0) , (0 , 1 , 0 , 1) , (1 , 1 , 0 , 1 ) , (1 , 1 , 0 , 2) , (0 , 0 , 1 , 0) , (0 , 0 , 1 , 1) , (1 , 0 , 1 , 1) , (1 , 0 , 1 , 2) , (0 , 1 , 1 , 1) , (0 , 1 , 1 , 2) , (1 , 1 , 1 , 1) , (1 , 1 , 1 , 2) , (1 , 1 , 1 , 3) o and h ead H ( ϑ ) 6∈ Γ. 1. T ( ϑ ) = (0 , 0 , 0 , 0). H ( ϑ ) (0,0,0,0) (0,0,0,1) (1,0,0,0) (0,1,0,0) (0,1,0,1) (0,0,1,0) w ( ϑ ) 0 -1 1 1 0 1 2. T ( ϑ ) = (0 , 0 , 0 , 1). H ( ϑ ) (0,0,0,2) (0,0,0,3) (1,0,0,1) (1,0,0,2) (0,1,0,2) (0,1,0,3) (1,1,0,1) w ( ϑ ) -3 -4 -1 -2 -2 -3 0 H ( ϑ ) (1,1,0,2) (0,0,1,1) (0,0,1,2) (1,0,1,1) (0,1,1,1) (0,1,1,2) (1,1,1,1) w ( ϑ ) -1 -1 -2 0 0 -1 1 3. T ( ϑ ) = (1 , 0 , 0 , 0). H ( ϑ ) (0,0,0,0) (0,1,0,0) w ( ϑ ) 1 2 4. T ( ϑ ) = (1 , 0 , 0 , 1). H ( ϑ ) (0,0,0,1) (0,0,0,2) (1,0,0,0) (1,0,0,1) (0,1,0,1) w ( ϑ ) -1 -2 1 0 0 H ( ϑ ) (0,1,0,2) (1,1,0,1) (0,0,1,0) (0,0,1,1) (0,1,1,1) w ( ϑ ) -1 1 1 0 1 5. T ( ϑ ) = (0 , 1 , 0 , 0). H ( ϑ ) (0,0,0,0) (0,1,0,0) w ( ϑ ) 1 2 6. T ( ϑ ) = (0 , 1 , 0 , 1). H ( ϑ ) (0,0,0,1) (0,0,0,2) (1,0,0,0) (1,0,0,1) (0,1,0,1) w ( ϑ ) -1 -2 1 0 0 H ( ϑ ) (0,1,0,2) (1,1,0,1) (0,0,1,0) (0,0,1,1) (0,1,1,1) w ( ϑ ) -1 1 1 0 1 22 7. T ( ϑ ) = (1 , 1 , 0 , 1). H ( ϑ ) (0,0,0,0) (0,0,0,1) (1,0,0,0) (0,1,0,0) (0,1,0,1) (0,0,1,0) w ( ϑ ) 1 0 2 2 1 2 8. T ( ϑ ) = (1 , 1 , 0 , 2). H ( ϑ ) (0,0,0,2) (0,0,0,3) (1,0,0,1) (1,0,0,2) (0,1,0,2) (0,1,0,3) (1,1,0,1) w ( ϑ ) -2 -3 0 -1 -1 -2 1 H ( ϑ ) (1,1,0,2) (0,0,1,1) (0,0,1,2) (1,0,1,1) (0,1,1,1) (0,1,1,2) (1,1,1,1) w ( ϑ ) 0 0 -1 1 1 0 2 9. T ( ϑ ) = (0 , 0 , 1 , 0). H ( ϑ ) (0,0,0,0) (0,1,0,0) w ( ϑ ) 1 2 10. T ( ϑ ) = (0 , 0 , 1 , 1). H ( ϑ ) (0,0,0,1) (0,0,0,2) (1,0,0,0) (1,0,0,1) (0,1,0,1) w ( ϑ ) -1 -2 1 0 0 H ( ϑ ) (0,1,0,2) (1,1,0,1) (0,0,1,0) (0,0,1,1) (0,1,1,1) w ( ϑ ) -1 1 1 0 1 11. T ( ϑ ) = (1 , 0 , 1 , 1). H ( ϑ ) (0,0,0,0) (0,0,0,1) (1,0,0,0) (0,1,0,0) (0,1,0,1) (0,0,1,0) w ( ϑ ) 1 0 2 2 1 2 12. T ( ϑ ) = (1 , 0 , 1 , 2). H ( ϑ ) (0,0,0,2) (0,0,0,3) (1,0,0,1) (1,0,0,2) (0,1,0,2) (0,1,0,3) (1,1,0,1) w ( ϑ ) -2 -3 0 -1 -1 -2 1 H ( ϑ ) (1,1,0,2) (0,0,1,1) (0,0,1,2) (1,0,1,1) (0,1,1,1) (0,1,1,2) (1,1,1,1) w ( ϑ ) 0 0 -1 1 1 0 2 13. T ( ϑ ) = (0 , 1 , 1 , 1). H ( ϑ ) (0,0,0,0) (0,0,0,1) (1,0,0,0) (0,1,0,0) (0,1,0,1) (0,0,1,0) w ( ϑ ) 1 0 2 2 1 2 14. T ( ϑ ) = (0 , 1 , 1 , 2). H ( ϑ ) (0,0,0,2) (0,0,0,3) (1,0,0,1) (1,0,0,2) (0,1,0,2) (0,1,0,3) (1,1,0,1) w ( ϑ ) -2 -3 0 -1 -1 -2 1 H ( ϑ ) (1,1,0,2) (0,0,1,1) (0,0,1,2) (1,0,1,1) (0,1,1,1) (0,1,1,2) (1,1,1,1) w ( ϑ ) 0 0 -1 1 1 0 2 23 15. T ( ϑ ) = (1 , 1 , 1 , 1). H ( ϑ ) (0,0,0,0) (0,1,0,0) w ( ϑ ) 2 3 16. T ( ϑ ) = (1 , 1 , 1 , 2). H ( ϑ ) (0,0,0,1) (0,0,0,2) (1,0,0,0) (1,0,0,1) (0,1,0,1) w ( ϑ ) 0 -1 2 1 1 H ( ϑ ) (0,1,0,2) (1,1,0,1) (0,0,1,0) (0,0,1,1) (0,1,1,1) w ( ϑ ) 0 2 2 1 2 17. T ( ϑ ) = (1 , 1 , 1 , 3). H ( ϑ ) (0,0,0,3) (0,0,0,4) (1,0,0,2) (1,0,0,3) (0,1,0,3) (0,1,0,4) (1,1,0,2) (1,1,0,3) w ( ϑ ) -3 -4 -1 -2 -2 -3 0 -1 H ( ϑ ) (0,0,1,2) (0,0,1,3) (1,0,1,1) (1,0,1,2) (0,1,1,2) (0,1,1,3) (1,1,1,1) (1,1,1,2) w ( ϑ ) -1 -2 1 0 0 -1 2 1 App endix B: T he Pro of of L emma 6 Pro of: The pro ofs of Lemma 6 (i) and (ii) are con tained in the pr o of of Lemma 6 (iii), so we only fo cus on the pro of for (iii). F urtherm ore, the pro of for the case P j +1 (2) = 1 and ω = − 2 is con tained in that for the case P j +1 (2) = 0 and ω = − 2, th u s w e alw ays assume that P j +1 (2) = 0 and ω = − 2 in the sequel. F or the same reason as in Example 1, without loss of generalit y , we can also assume that the in teger q is large enough. Let ϑ i denote th e arc with the tail P i and head P i +1 for eac h i ∈ { 0 , 1 , · · · , q − 1 } . Since P j +1 (2) = 0 and ω = − 2, b y P j = (0 , 0 , 0 , 0) and App endix A, we hav e P j +1 ∈ (0 , 0 , 0 , 0) , (0 , 0 , 0 , 1) , (0 , 0 , 1 , 0) , (1 , 0 , 0 , 0) . If P j +1 = (0 , 0 , 0 , 0), then w ( ϑ j ) = 0. By a similar analysis as in (15), we ha ve w ( ϑ j +1 ) ≥ − 1. Consequently , P j W P j +2 has the form as (0 , 0 , 0 , 0) (0 , − 2) − − − − → (0 , 0 , 0 , 0) (0 , − 1) − − − − → (Φ1) . (29) F or P j +1 ∈ (0 , 0 , 0 , 1) , (0 , 0 , 1 , 0) , (1 , 0 , 0 , 0) , b y a similar analysis P j W P j +2 has other three p ossible forms as b elo w. (0 , 0 , 0 , 0) (0 , − 2) − − − − → (0 , 0 , 0 , 1) (0 , 0) − − − → (Φ2) (0 , 0 , 1 , 0) (0 , − 2) − − − − → (Φ3) (1 , 0 , 0 , 0) (0 , − 2) − − − − → (Φ4) . 24 In th e case (Φ1), P j +2 (2) = 0 and then b y App end ix A, we ha ve P j +2 ∈ n (0 , 0 , 0 , 0) , (0 , 0 , 0 , 1) , (0 , 0 , 1 , 0) , (1 , 0 , 0 , 0) o . Since the w eights of the arcs with the tail P j +1 and heads (0 , 0 , 0 , 0) , (0 , 0 , 0 , 1), (0 , 0 , 1 , 0) , (1 , 0 , 0 , 0) are 0, − 1, 1, 1, resp ectiv ely , there are four p ossible f orm s for P j W P j +3 as (0 , 0 , 0 , 0) (0 , − 2) − − − − → (0 , 0 , 0 , 0) (0 , − 1) − − − − → (0 , 0 , 0 , 0) (0 , 0) − − − → (Φ1 . 1) (0 , 0 , 0 , 1) (0 , 1) − − − → (Φ1 . 2) (0 , 0 , 1 , 0) (0 , − 1) − − − − → (Φ1 . 3) (1 , 0 , 0 , 0) (0 , − 1) − − − − → (Φ1 . 4) . F or the case (Φ1 . 1), P j +3 (2) = 0 and w ( ϑ j +2 ) ≥ 0. So P j +3 ∈ (0 , 0 , 0 , 0) , (1 , 0 , 0 , 0) , (0 , 0 , 1 , 0) b y App end ix A. When P j +3 = (0 , 0 , 0 , 0), we hav e w ( ϑ j +2 ) = 0 and w ( ϑ j +3 ) ≥ 0 + 1 − w ( ϑ j +2 ) = 1. By Example 1, in the case (Φ1 . 1) and P j +3 = (0 , 0 , 0 , 0), all ve r tices P l ( j +1 ≤ l ≤ q ) o ccurrin g in the w alk W are cont ained in the set { (0 , 0 , 0 , 0) , (1 , 0 , 0 , 0) , (0 , 1 , 0 , 0) , (0 , 0 , 1 , 0) } , wh ic h is a subset of S 1 . When P j +3 = (1 , 0 , 0 , 0), b y a similar analysis P j +3 W P j +5 has the form (1 , 0 , 0 , 0) (0 , 0) − − − → (0 , 0 , 0 , 0) (0 , 0) − − − → . When P j +3 = (0 , 0 , 1 , 0), P j +3 W P j +7 has thr ee p ossible forms (0 , 0 , 1 , 0) (0 , 0) − − − → (0 , 0 , 0 , 0) (1 , 0) − − − → (0 , 1 , 0 , 1) (0 , 1) − − − → (0 , 0 , 1 , 0) (0 , 1) − − − → (1 , 0 , 0 , 0) (0 , 1) − − − → (0 , 1 , 0 , 0) (0 , 0) − − − → (0 , 0 , 0 , 0) (0 , 0) − − − → . Therefore, for the case (Φ1 . 1), (0 , 0 , 0 , 0) (0 , 0) − − − → has an SBS as (0 , 0 , 0 , 0) (0 , 0) − − − → (0 , 0 , 0 , 0) (0 , 1) − − − → (0 , 0 , 1 , 0) (0 , 1) − − − → (0 , 0 , 0 , 0) (1 , 1) − − − → (0 , 1 , 0 , 0) (0 , 1) − − − → (0 , 0 , 0 , 0) (0 , 1) − − − → (1 , 0 , 0 , 0) (0 , 1) − − − → (0 , 0 , 0 , 0) (0 , 1) − − − → (0 , 0 , 1 , 0) (0 , 0) − − − → (0 , 0 , 0 , 0) (1 , 0) − − − → (0 , 1 , 0 , 1) (0 , 1) − − − → (0 , 0 , 1 , 0) (0 , 1) − − − → (1 , 0 , 0 , 0) (0 , 1) − − − → (0 , 1 , 0 , 0) (0 , 0) − − − → (0 , 0 , 0 , 0) (0 , 0) − − − → (1 , 0 , 0 , 0) (0 , 0) − − − → (0 , 0 , 0 , 0) (0 , 0) − − − → , (30) in which all vertices and arcs in P j +2 W P q ha ve o ccurred for the case P j +2 = (0 , 0 , 0 , 0). Th us, all v ertices P l ( j + 1 ≤ l ≤ q ) o ccurring in the w alk W are con tained in the set S 1 defined b y 25 (21). F urth ermore, b y (30), all w alks w ith the f orm (0 , 0 , 0 , 0) ( η, ω ) − − − → for η ∈ { 0 , 1 } and ω ∈ { 0 , 1 } ha ve o ccurred in (30). T h is finishes th e p ro of of Lemma 6 (i). F or the case (Φ1 . 2), by Ap p endix A, we h av e (0 , 0 , 0 , 1) (0 , 1) − − − → O , i.e., q = j + 2 and P j +2 = P q . F or the case (Φ1 . 3), P j +2 W P j +6 has five p ossible forms as (0 , 0 , 1 , 0) (0 , − 1) − − − − → (0 , 0 , 0 , 0) (1 , − 1) − − − − → (0 , 1 , 0 , 0) (0 , − 1) − − − − → (0 , 0 , 0 , 0) (0 , − 1) − − − − → (Φ1 . 3 . 1) (0 , 1 , 0 , 1) (0 , 0) − − − → (0 , 0 , 1 , 0) (0 , 0) − − − → (Φ1 . 3 . 2) (0 , 0 , 1 , 1) (0 , 1) − − − → (Φ1 . 3 . 3) (1 , 0 , 0 , 0) (0 , 0) − − − → (Φ1 . 3 . 4) (1 , 0 , 0 , 1) (0 , 1) − − − → (Φ1 . 3 . 5) . (31) The w alks (0 , 0 , 1 , 0) (0 , 0) − − − → in (Φ1 . 3 . 2 ) and (1 , 0 , 0 , 0) (0 , 0) − − − → in (Φ1 . 3 . 4 ) ha ve o ccurred in (30). W e need to fur th er analyze the cases (Φ1 . 3 . 3) and (Φ1 . 3 . 5). By App endix A, (0 , 0 , 1 , 1) (0 , 1) − − − → has an SBS as (0 , 0 , 1 , 1) (0 , 1) − − − → (0 , 0 , 1 , 0) (1 , 1) − − − → (0 , 1 , 0 , 0) (1 , 0) − − − → (0 , 1 , 0 , 0) (0 , − 1) − − − − → (0 , 0 , 0 , 0) (0 , − 1) − − − − → (1 , 0 , 0 , 0) (1 , 1) − − − → (0 , 1 , 0 , 0) (0 , 0) − − − → (0 , 0 , 0 , 0) (0 , 0) − − − → (32) for the case (Φ1 . 3 . 3) and (1 , 0 , 0 , 1) (0 , 1) − − − → has an S BS as (1 , 0 , 0 , 1) (0 , 1) − − − → (0 , 0 , 1 , 0) (0 , 1) − − − → (0 , 0 , 0 , 0) (1 , 1) − − − → (1 , 0 , 0 , 0) (0 , 1) − − − → (0 , 0 , 0 , 0) (0 , 1) − − − → (33) for the case (Φ1 . 3 . 5). F or the case (Φ1 . 4), P j +2 W P j +4 is giv en by (1 , 0 , 0 , 0) (0 , − 1) − − − − → (0 , 0 , 0 , 0) (0 , − 1) − − − − → . Notice that the walk (0 , 0 , 0 , 0) (0 , − 1) − − − − → in (Φ1 . 3 . 1), (Φ1 . 3 . 3 ) and (Φ1 . 4) h as occurr ed as P j +1 W P j +2 in (29). Therefore, b y the ab o ve analysis for (Φ1 . 1)-(Φ1 . 4) and Lemma 6 (i), in the case that P j W P j +2 has the form as (29), all v ertices P l ( j + 1 ≤ l ≤ q ) o ccurring in th e w alk W are con tained in the set S 2 defined b y (22). F u rthermore, the w alks (0 , 0 , 0 , 0) ( η, − 1) − − − − → for η ∈ { 0 , 1 } hav e o ccurr ed in (31). T his finish es the p ro of of Lemma 6 (ii). 26 F or the case (Φ2), (0 , 0 , 0 , 1) (0 , 0) − − − → (1 , 0 , 1 , 1) has an SBS as (0 , 0 , 0 , 1) (0 , 0) − − − → (1 , 0 , 1 , 1) (0 , 1) − − − → (0 , 0 , 0 , 0) (1 , 1) − − − → (0 , 0 , 1 , 0) (1 , 0) − − − → (0 , 1 , 0 , 0) (1 , − 1) − − − − → (0 , 1 , 0 , 0) (0 , − 2) − − − − → (0 , 0 , 0 , 0) (0 , − 2) − − − − → (1 , 0 , 0 , 0) (1 , 0) − − − → (0 , 1 , 0 , 0) (0 , − 1) − − − − → (0 , 0 , 0 , 0) (0 , − 1) − − − − → . F or the case (Φ3), P j +1 W P j +5 has six p ossible f orm s as (0 , 0 , 1 , 0) (0 , − 2) − − − − → (0 , 0 , 0 , 0) (1 , − 2) − − − − → (0 , 1 , 0 , 0) (0 , − 2) − − − − → (0 , 0 , 0 , 0) (0 , − 2) − − − − → (Φ 3 . 1) (0 , 1 , 0 , 1) (0 , − 1) − − − − → (0 , 0 , 0 , 1) (0 , 1) − − − → (Φ3 . 2) (0 , 0 , 1 , 0) (0 , − 1) − − − − → (Φ3 . 3) (0 , 0 , 1 , 1) (0 , 0) − − − → (Φ 3 . 4) (1 , 0 , 0 , 0) (0 , − 1) − − − − → (Φ3 . 5) (1 , 0 , 0 , 1) (0 , 0) − − − → (Φ3 . 6) . (34) The wa lk (0 , 0 , 0 , 1) (0 , 1) − − − → in (Φ3 . 2) has o ccurred as P j +2 W P j +3 in (Φ1 . 2). F or the case (Φ3 . 3), since the segmen t P j +4 W P j +5 has the form (0 , 0 , 1 , 0) (0 , − 1) − − − − → , the segmen t P j +4 W P j +6 has the form (0 , 0 , 1 , 0) (0 , − 1) − − − − → (0 , 0 , 0 , 0) (1 , − 1) − − − − → . By Lemm a 6 (ii), for the cases (Φ3 . 2) and (Φ3 . 3), all v ertices in W are con tained in the set S 2 . F or the case (Φ3 . 4), (0 , 0 , 1 , 1) (0 , 0) − − − → has an S BS as (0 , 0 , 1 , 1) (0 , 0) − − − → (0 , 0 , 1 , 0) (1 , 0) − − − → (0 , 1 , 0 , 0) (1 , − 1) − − − − → (0 , 1 , 0 , 0) (0 , − 2) − − − − → (0 , 0 , 0 , 0) (0 , − 2) − − − − → (Φ3 . 4 . 1 ) (0 , 0 , 1 , 1) (1 , 1) − − − → (1 , 1 , 0 , 1) (1 , 1) − − − → (Φ3 . 4 . 2) (0 , 1 , 1 , 1) (1 , 1) − − − → (Φ3 . 4 . 3) (1 , 0 , 0 , 0) (1 , 0) − − − → (0 , 1 , 0 , 0) (0 , − 1) − − − − → (0 , 0 , 0 , 0) (0 , − 1) − − − − → (Φ3 . 4 . 4) (1 , 0 , 0 , 1) (1 , 1) − − − → (0 , 1 , 1 , 1) (0 , 1) − − − → (Φ3 . 4 . 5) (1 , 1 , 0 , 1) (0 , 1) − − − → (Φ3 . 4 . 6) . F or the case (Φ3 . 4 . 2) , (0 , 0 , 1 , 1) (1 , 1) − − − → has an S BS as (0 , 0 , 1 , 1) (1 , 1) − − − → (1 , 1 , 0 , 1) (1 , 1) − − − → (0 , 1 , 0 , 0) (0 , 0) − − − → (0 , 0 , 0 , 0) (0 , 0) − − − → (0 , 1 , 0 , 1) (0 , 1) − − − → and the wa lk (0 , 1 , 0 , 1) (0 , 1) − − − → has o ccurred in (30). F or the case (Φ3 . 4 . 3 ), P j +5 W P j +9 has three 27 p ossible forms as (0 , 0 , 1 , 1) (1 , 1) − − − → (0 , 1 , 1 , 1) (1 , 1) − − − → (0 , 1 , 0 , 0) (1 , 0) − − − → (0 , 1 , 0 , 0) (0 , − 1) − − − − → (Φ3 . 4 . 3 . 1) (0 , 1 , 0 , 1) (1 , 1) − − − → (0 , 1 , 1 , 1) (0 , 1) − − − → (Φ3 . 4 . 3 . 2 ) (1 , 1 , 0 , 1) (0 , 1) − − − → (Φ3 . 4 . 3 . 3 ) . Since the walk (0 , 1 , 0 , 0) (0 , − 1) − − − − → in th e case (Φ3 . 4 . 3 . 1) has o ccurred in the case (Φ1 . 3 . 1) as (31), w e need to further an alyze the cases (Φ3 . 4 . 3 . 2) and (Φ3 . 4 . 3 . 3). (0 , 1 , 1 , 1) (0 , 1) − − − → has an S BS as (0 , 1 , 1 , 1) (0 , 1) − − − → (0 , 0 , 0 , 0) (1 , 1) − − − → (0 , 0 , 1 , 0) (1 , 0) − − − → (0 , 1 , 0 , 0) (1 , − 1) − − − − → (0 , 1 , 0 , 0) (0 , − 2) − − − − → (0 , 0 , 0 , 0) (0 , − 2) − − − − → (1 , 0 , 0 , 0) (1 , 0) − − − → (0 , 1 , 0 , 0) (0 , − 1) − − − − → (0 , 0 , 0 , 0) (0 , − 1) − − − − → (35) for (Φ3 . 4 . 3 . 2), and (1 , 1 , 0 , 1) (0 , 1) − − − → has an S BS as (1 , 1 , 0 , 1) (0 , 1) − − − → (0 , 0 , 0 , 0) (0 , 1) − − − → (0 , 0 , 1 , 0) (0 , 0) − − − → (0 , 0 , 0 , 0) (1 , 0) − − − → (1 , 0 , 0 , 0) (0 , 0) − − − → (0 , 0 , 0 , 0) (0 , 0) − − − → (36) for (Φ3 . 4 . 3 . 3). Notice that the w alk (0 , 0 , 0 , 0) (0 , − 1) − − − − → in (Φ3 . 4 . 4) has o ccurred in (Φ1) and the wa lks (0 , 1 , 1 , 1) (0 , 1) − − − → in (Φ3 . 4 . 5 ) and (1 , 1 , 0 , 1) (0 , 1) − − − → in (Φ3 . 4 . 6) h a v e b een analyzed in (35) and (36), resp ectiv ely . F or the case (Φ3 . 5), P j +4 W P j +6 has the form as (1 , 0 , 0 , 0) (0 , − 1) − − − − → (0 , 0 , 0 , 0) (0 , − 1) − − − − → and for the case (Φ3 . 6), (1 , 0 , 0 , 1) (0 , 0) − − − → has an S BS as (1 , 0 , 0 , 1) (0 , 0) − − − → (0 , 0 , 1 , 0) (0 , 0) − − − → (0 , 0 , 0 , 0) (1 , 0) − − − → (0 , 0 , 1 , 1) (0 , 1) − − − → (1 , 0 , 0 , 0) (0 , 0) − − − → (0 , 0 , 0 , 0) (0 , 0) − − − → (1 , 0 , 0 , 1) (0 , 1) − − − → . Notice that the walks (0 , 0 , 1 , 1) (0 , 1) − − − → and (1 , 0 , 0 , 1) (0 , 1) − − − → hav e b een analyzed in (32) and (33), resp ectiv ely . F or the case (Φ4), the segmen t P j +1 W P j +3 has the form (1 , 0 , 0 , 0) (0 , − 2) − − − − → (0 , 0 , 0 , 0) (0 , − 2) − − − − → . Notice that the walk (0 , 0 , 0 , 0) (0 , − 2) − − − − → in the cases (Φ2), (Φ3 . 1), (Φ3 . 4 . 1), (Φ3 . 4 . 3 . 2), (Φ3 . 4 . 5), and (Φ4) has o ccurred as P j W P j +1 . Th erefore, co mbining the abov e analysis for the cases (Φ2)- (Φ4) and by Lemma 6 (i), (ii), all v ertices P l ( j + 1 ≤ l ≤ q ) occurr ing in the w alk W a re 28 con tained in the set S 3 . The p ro of for th e case η = 1 and ω = − 2 is con tained in the analysis of th e case (Φ3) in (34). This fi nishes the pro of of Lemma 6 (iii). 29
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