New Families of Triple Error Correcting Codes with BCH Parameters

New F amilies of T riple Error Correcting Co des with B CH P arameters Carl Brack en Sc ho o l of Mathem ati ca l Sciences Universit y Coll ege Dubl in Ireland No vem ber 21, 2021 Abstract Disco v ered by Bose, Chaud huri a nd Ho cquenghem [1], [4], the BCH family of error correcting co d es are one of the most s tudied fam- ilies in c o ding theory . They are a lso among the b est p erforming co des, particularly when the n umber of errors b eing corrected is small rela- tiv e to the co de length. In this article w e consider b inary co des with minim um distance of 7. W e construct new families of co d es with these BCH parameters via a generalisation of the Kasami-W elc h T heorem. 1 In tro d uction Let L = GF (2 n ) for some o dd n and let L ∗ denote the non zero elemen ts o f L . Also, let f ( x ) and g ( x ) b e t w o mappings from L to itself. W e assume the functions are c hosen suc h that f (0) = g (0) = 0. W e can construct the parit y 1 c hec k matrix H of an error correcting co de C as follows. H =     .... x .... .... f ( x ) .... .... g ( x ) ....     The co de C is defined as the n ullspace of H . Eac h column is a binary v ector of length 3 n comp osed of the binary represen tations of three elemen ts of L ∗ with resp ect to some c hosen basis. This matrix is a 3 n b y 2 n − 1 array and hence C has parameters [2 n − 1 , 2 n − 3 n − 1 , d ]. This means that C is a co de of dimension 2 n − 3 n − 1 and minimum Hamming distance d b et w ee n an y pair of ve ctors (usually called words). The n um b er of errors a co de can correct is giv en b y e = d − 1 2 . The co de generated by t he ro ws of H is called the dual o f C a nd is denoted b y C ⊥ . W e can determine the w eigh ts of the w ords in C ⊥ b y computing the generalised F o urier tr ansform of the pair of functions { f ( x ) , g ( x ) } . W e define this transform as F w ( a, b, c ) = X x ∈ GF (2 n ) ( − 1) T r ( ax + bf ( x )+ cg ( x )) with b, c ∈ L ∗ and a ∈ L . In order to demonstrate that the minim um distance of C is 7 w e w ill req uire this transform to b e limited to fiv e sp ecifie d v alues. W e will also require at least o ne of our functions to b e Almost P erfect Nonlinear (APN). The map h is said to b e APN on L if the n umber of solutions in L of the equation h ( x + q ) − h ( x ) = p is at most 2, fo r all p ∈ L and q ∈ L ∗ . An APN function can b e used to construct a co de with double error correcting BCH para meters. F or an in tro duction to the connections b et w een F ourier transforms, APN functions and BCH co des, we recommend pag es 1037-1039 of [5]. The w eights in C ⊥ are giv en by w = 1 2 (2 n − V ), where V ranges o v er the v alues of F w ( a, b, c ). If at least one of the pair { f ( x ) , g ( x ) } is APN then t he co de constructed with this pair will ha v e a minim um distance of at least 5 as it w ill b e the su b co de 2 of a co de with double error correcting BCH parameters ([5] page 1037). The MacWilliams iden tities ([5] page 88) are a set of equations that allow us to compute the w eigh t distribution of a linear co de from the we ight distribution of its dual. Let A w b e the nu m b er of co dew ords in C with w eight w . As the distance in C is at least 5 w e ha ve, A 4 = A 3 = A 2 = A 1 = 0 and A 0 = 1. If we demonstrate that the w eights in C ⊥ tak e the same five v alues as those o ccuring in the dual of the triple error corr ecting BCH code, then w e can use the MacWilliams iden tities to compute the multiplicit y of the w eights o ccuring in C ⊥ . As w e ha v e fiv e equations in fiv e unkno wns the m ultiplicity is exactly determined. There fore C ⊥ and hence C m ust hav e the same w eigh t distribution as the triple error correcting BCH co de. The w ords in the dual of the triple erro r c orrecting BCH tak e the fiv e w eigh ts 2 n − 1 − 2 n − 1 2 , 2 n − 1 + 2 n − 1 2 , 2 n − 1 − 2 n +1 2 , 2 n − 1 + 2 n +1 2 and 2 n − 1 . It follow s that if w e show that F w ( a, b, c ) is limited to t he fiv e v alues 0 , ± 2 n +1 2 , ± 2 n +3 2 for some pair o f f unctions with at least one being APN, then the co de constructed from this pair will ha v e a minim um distance of 7. 3 2 Kno w n F amilies of C o des The follo wing table lists the kno wn triple error correcting co des that can b e constructed with pairs of functions ov er GF (2 n ) via the metho d outlined ab ov e. f ( x ) , g ( x ) Conditions Ref. x 2 k +1 , x 2 2 k +1 g cd ( n, k ) = 1 [1], [4 ] Theorem 1 x 2 k +1 , x 2 3 k +1 g cd ( n, k ) = 1 [7] Theorem 1 x 2 t +1 , x 2 t +2 +3 n = 2 t + 1 [9] x 2 2 k − 2 k +1 , x 2 4 k − 2 3 k +2 2 k − 2 k +1 g cd ( n, k ) = 1 Theorem 2 . Next w e will demonstrate that the first tw o pairs of functions from the ab ov e table indeed allo w us to construct co des with a minim um distance of 7. W e do this by computing the generalised F ourier transforms of eac h pair. W e b eliev e that the results in this section are w ell kno wn to those working in the area but w e ha v e not seen a pro of in the litreture. The codes giv en b y the first family are a generalisation of the classic, also called primitiv e, BCH co des defined b y the pair of functions { x 3 , x 5 } . The second family generalises the non-BCH triple erro r correcting co des f r om [7] defined b y the pair of functions { x 2 t +1 , x 2 t − 1 +1 } when n = 2 t + 1 . Be fore w e pro ceed, we state the follo wing often used result, a pro of of whic h can b e f ound in [2 ]. 4 Lemma 1 L et L = GF (2 n ) a n d let p ( x ) = P d i =0 r i x 2 ki b e a p olynomial in L [ x ] with r i ∈ L and gcd( k , n ) = 1 . Then p ( x ) has at most 2 d solutions in L . Theorem 2 L et n b e o dd and k r elatively prime to n . The p air of functions { x 2 k +1 , g ( x ) } c onstruct a triple err or c orr e cting c o de with BCH p ar ameters whenever g ( x ) = x 2 2 k +1 or x 2 3 k +1 . Pro of: As x 2 k +1 is APN [6], w e can obtain the result b y sho wing that the g eneralised F ourier transform alwa ys ta k es one o f the fiv e v alues 0 , ± 2 n +1 2 or ± 2 n +3 2 . Let g ( x ) = x 2 tk +1 where t = 2 or 3. By definition, w e hav e F W ( a, b, c ) = X x ∈ L ( − 1) T r( ax + bx 2 k +1 + cx 2 tk +1 ) . W e let Q ( x ) = ax + bx 2 k +1 + cx 2 tk +1 and take the square of the transform to get, ( F W ( a, b, c )) 2 = X x ∈ L X y ∈ L ( − 1) T r( Q ( x )) ( − 1) T r( Q ( y )) . Replace y with x + u to obtain ( F w ( a, b, c )) 2 = X u ( − 1) T r ( Q ( u )) X x ( − 1) T r ( x ( L ( u ))) , where L ( u ) = bu 2 k + b 2 − k u 2 − k + cu 2 tk + c 2 − tk u 2 − tk . Next w e use the the fact that P x ( − 1) T r ( γ x ) is 2 n when γ = 0, and is 0 otherwise to obtain ( F w ( a, b, c )) 2 = 2 n X u ∈ K ( − 1) T r ( Q ( u )) where K is the k ernel o f L ( u ) . Consider χ ( u ) = ( − 1) T r ( Q ( u )) . It can b e easily demonstrated that χ a is a c haracter of K as χ ( u + v ) = χ ( u ) χ ( v ) . 5 No w using the fact that, for a c ha racter χ of a group H , P h ∈ H χ ( h ) = | H | if χ is the iden t ity c haracter and 0 otherwise (see [8], 6 2-63) w e see that ( F w ( a, b, c )) 2 = 0 or 2 n + s , where 2 s is the n umber o f solutions t o L ( u ) = 0. Raising L ( u ) b y 2 tk w e obtain the fo llowing equation b 2 tk u 2 ( t +1) k + b 2 ( t − 1) k u 2 ( t − 1) k + cu 2 2 tk + cu = 0 . Next w e apply Lemma 1 and hence bound s b y 2 s ≤ 2 4 when t = 2 and 2 s ≤ 2 3 when t = 3. As F w ( a, b, c ) m ust b e an in teger, in b oth cases w e hav e F w ( a, b, c ) = 0 , ± 2 n +1 2 or ± 2 n +3 2 and w e ar e done. 3 New F amilies o f Co des In this section we pro v e that the fourth pair of functions from the list yield a co de that is triple error correcting. Again, w e do this b y computing the generalised F ourier transform. T he pro of uses some of the tec hniques from Hans Dobb ertin’s pro o f of the Kasami-W elc h theorem [3]. Theorem 3 F or n o dd and k r elatively prime to n , the p air of functions { x 2 2 k − 2 k +1 , x 2 4 k − 2 3 k +2 2 k − 2 k +1 } on GF (2 n ) c onstruct a triple err or c orr e cting c o de with BC H p ar ame ters. Pro of: Again, as x 2 2 k − 2 k +1 is APN [3 ], w e ac hiev e the result b y computing the gen- eralised F ourier transform. By definition, F w ( a, b, c ) = X x ∈ GF (2 n ) ( − 1) T r ( ax + bx 2 2 k − 2 k +1 + cx 2 4 k − 2 3 k +2 2 k − 2 k +1 ) . Replacing x with x 2 k +1 (a p erm utation) we obtain F w ( a, b, c ) = X x ∈ GF (2 n ) ( − 1) T r ( ax 2 k +1 + bx 2 3 k +1 + cx 2 5 k +1 ) . 6 Letting Q ( x ) = ax 2 k +1 + bx 2 3 k +1 + cx 2 5 k +1 and squaring giv es, ( F w ( a, b, c )) 2 = X x X y ( − 1) T r ( Q ( x )+ Q ( y )) . Let y = x + u to obtain ( F w ( a, b, c )) 2 = X u ( − 1) T r ( Q ( u )) X x ( − 1) T r ( x ( L ( u ))) where L ( u ) = au 2 k + a 2 − k u 2 − k + bu 2 3 k + b 2 − 3 k u 2 − 3 k + cu 2 5 k + c 2 − 5 k u 2 − 5 k . As P x ( − 1) T r ( γ x ) is 2 n when γ = 0 , and is 0 otherwise, we hav e ( F w ( a, b, c )) 2 = 2 n X u ∈ K ( − 1) T r ( Q ( u )) where K is the k ernel of L ( u ) . Let K = S 0 ∪ S 1 , where S 0 = { u ∈ K : T r ( Q ( u )) = 0 } and S 1 = { u ∈ K : T r ( Q ( u )) = 1 } . W e no w see that F w ( a, b, c ) = ± p 2 n ( | S 0 | − | S 1 | . Therefore, w e need o nly sho w tha t | S 0 | − | S 1 | < 32 for the result to fo llo w. T o this end w e let G ( u ) = au 2 k +1 + bu 2 3 k +1 + b 2 − k u 2 2 k +2 − k + b 2 − 2 k u 2 k +2 − 2 k + cu 2 5 k +1 + c 2 − k u 2 4 k +2 − k + c 2 − 2 k u 2 3 k +2 − 2 k + c 2 − 3 k u 2 2 k +2 − 3 k + c 2 − 4 k u 2 k +2 − 4 k . Note that G ( u ) + G ( u ) 2 − k = uL ( u ) . 7 Therefore u ∈ K if and only if G ( u ) = 0 or 1 (as g cd ( k , n ) = 1). F urthermore, u ∈ S 0 if a nd only if G ( u ) = 0 . Next let χ ( u ) = ( − 1) T r ( au 2 k +1 + bu 2 3 k +1 + cu 2 5 k +1 ) . A s in Lemma 2 , it can b e sho wn that χ a is a character of K a nd hence | S 0 | − | S 1 | = 0 or | K | . As | S 0 | − | S 1 | = X u ∈ K χ ( u ) it follo ws that if F w ( a ) 6 = 0 then K = S 0 . That is w e can assume L ( u ) = 0 has the same solution set as G ( u ) = 0 and that | S 1 | has no solutions when F w ( a, b, c ) 6 = 0. F urthermore, as F w ( a, b, c ) ∈ { 0 , ± p 2 n ( | S 0 | ) } w e can say that | S 0 | m ust b e an o dd p o w er of t w o . It remains to sho w that G ( u ) = 0 has at most 16 solutions when K = S 0 . F or the sake of con t radiction w e assume G ( u ) = 0 has 32 solutions a nd that they form an additiv e group. Now apply the iden tit y ( u + v )( v G ( u ) + uG ( v )) + uv ( G ( u + v )) = 0 (1) for three solutions u, v and u + v with u 6 = v and v 6 = 0. W e then rearrange to obtain the follo wing expression b 2 − k ( u 2 2 k v + uv 2 2 k )( u 2 − k v + uv 2 − k ) + b 2 − 2 k ( u 2 k v + uv 2 k )( u 2 − 2 k v + uv 2 − 2 k )+ c 2 − k ( u 2 4 k v + uv 2 4 k )( u 2 − k v + uv 2 − k ) + c 2 − 2 k ( u 2 3 k v + uv 2 3 k )( u 2 − 2 k v + uv 2 − 2 k )+ c 2 − 3 k ( u 2 2 k v + uv 2 2 k )( u 2 − 3 k v + uv 2 − 3 k ) + c 2 − 4 k ( u 2 k v + uv 2 k )( u 2 − 4 k v + uv 2 − 4 k ) = 0 . Under our assumption the ab o v e equation has 32 solutio ns in u , for a fixed (non zero) solution v . Next let u = v w and divide by v 2 to obtain b 2 − k v 2 2 k +2 − k ( w + w 2 2 k )( w + w 2 − k ) + b 2 − 2 k v 2 k +2 − 2 k ( w + w 2 k )( w + w 2 − 2 k )+ c 2 − k v 2 4 k +2 − k ( w + w 2 4 k )( w + w 2 − k ) + c 2 − 2 k v 2 3 k +2 − 2 k ( w + w 2 3 k )( w + w 2 − 2 k )+ c 2 − 3 k v 2 2 k +2 − 3 k ( w + w 2 2 k )( w + w 2 − 3 k ) + c 2 − 4 k v 2 k +2 − 4 k ( w + w 2 k )( w + w 2 − 4 k ) = 0 . 8 No w letting w + w 2 − k = r , w e obtain the f ollo wing equation, with 16 solutions in r whic h for m a n additiv e gr o up as w + w 2 − k is 2 − to − 1 and linear. b 2 − k v 2 2 k +2 − k ( r 2 k + r 2 2 k )( r ) + b 2 − 2 k v 2 k +2 − 2 k ( r 2 − k )( r + r 2 − k )+ c 2 − k v 2 4 k +2 − k ( r 2 k + r 2 2 k + r 2 3 k + r 2 4 k )( r )+ c 2 − 2 k v 2 3 k +2 − 2 k ( r 2 k + r 2 2 k + r 2 3 k )( r + r 2 − k )+ c 2 − 3 k v 2 2 k +2 − 3 k ( r 2 k + r 2 2 k )( r + r 2 − k + r 2 − 2 k )+ c 2 − 4 k v 2 k +2 − 4 k ( r 2 k )( r + r 2 − k + r 2 − 2 k + r 2 − 3 k ) = 0 . Rearranging giv es Ar 2 k +1 + B r 2 2 k +1 + B 2 − k r 2 k +2 − k + C r 2 3 k +1 + C 2 − k r 2 2 k +2 − k + C 2 − 2 k r 2 k +2 − 2 k + D r 2 4 k +1 + D 2 − k r 2 3 k +2 − k + D 2 − 2 k r 2 2 k +2 − 2 k + D 2 − 3 k r 2 k +2 − 3 k = 0 , where A, B , C and D are functions of b, c and v , that is, they are fixed elemen ts of the field. No w, re-apply the iden tity (1 ) for the ab ov e equation with three solutions r, s and r + s (where s will b e a fixed non zero solution) to obtain B 2 − k ( r 2 k s + r s 2 k )( r 2 − k s + r s 2 − k ) + C 2 − k ( r 2 2 k s + r s 2 2 k )( r 2 − k s + r s 2 − k )+ C 2 − 2 k ( r 2 k s + r s 2 k )( r 2 − 2 k s + r s 2 − 2 k ) + D 2 − k ( r 2 3 k s + r s 2 3 k )( r 2 − k s + r s 2 − k )+ D 2 − 2 k ( r 2 2 k s + r s 2 2 k )( r 2 − 2 k s + r s 2 − 2 k ) + D 2 − 3 k ( r 2 k s + r s 2 k )( r 2 − 3 k s + r s 2 − 3 k ) = 0 . Next let r = st , to giv e the follow ing equation whic h will hav e 16 solutions in t : B 2 − k s 2 k +2 − k ( t + t 2 k )( t + t 2 − k ) + C 2 − k s 2 2 k +2 − k ( t + t 2 2 k )( t + t 2 − k )+ C 2 − 2 k s 2 k +2 − 2 k ( t + t 2 k )( t + t 2 − 2 k ) + D 2 − k s 2 3 k +2 − k ( t + t 2 3 k )( t + t 2 − k )+ D 2 − 2 k s 2 2 k +2 − 2 k ( t + t 2 2 k )( t + t 2 − 2 k ) + D 2 − 3 k s 2 k +2 − 3 k ( t + t 2 k )( t + t 2 − 3 k ) = 0 . 9 No w let t + t 2 − k = z to obtain B 2 − k s 2 k +2 − k ( z 2 k )( z ) + C 2 − k s 2 2 k +2 − k ( z 2 k + z 2 2 k )( z )+ C 2 − 2 k s 2 k +2 − 2 k ( z 2 k )( z + z 2 − k ) + D 2 − k s 2 3 k +2 − k ( z 2 k + z 2 2 k + z 2 3 k )( z )+ D 2 − 2 k s 2 2 k +2 − 2 k ( z 2 k + z 2 2 k )( z + z 2 − k ) + D 2 − 3 k s 2 k +2 − 3 k ( z 2 k )( z + z 2 − k + z 2 − 2 k ) = 0 . The ab o v e equation will ha ve eight solutions in z , whic h will again form an additiv e gr oup. No w apply the iden tit y (1) fo r three solutions z , p and z + p to get E ( z 2 k p + z p 2 k )( z 2 − k p + z p 2 − k ) + F ( z 2 2 k p + z p 2 2 k )( z 2 − k p + z p 2 − k )+ F 2 − k ( z 2 k p + z p 2 k )( z 2 − 2 k z + z p 2 − 2 k ) = 0 , where E and F are fixed elemen ts of the field. Letting q = z p and dividing b y p 2 yields E p 2 k +2 − k ( q + q 2 k )( q + q 2 − k ) + F p 2 2 k +2 − k ( q + q 2 2 k )( q + q 2 − k )+ F 2 − k p 2 k +2 − 2 k ( q + q 2 k )( q + q 2 − 2 k ) = 0 . No w let q + q 2 − k = d , to obtain the following equation with four solutions in d : E p 2 k +2 − k ( d 2 k )( d ) + F p 2 2 k +2 − k ( d 2 k + d 2 2 k )( d ) + F 2 − k p 2 k +2 − 2 k ( d 2 k )( d + d 2 − k ) = 0 . Apply the iden tit y ( 1 ) once more, for three solutio ns d, e and d + e to obtain F 2 − k p 2 k +2 − 2 k ( d 2 k e + de 2 k )( d 2 − k e + de 2 − k ) = 0 . It needs to b e v erified at this p oin t that the constant term F 2 − k p 2 k +2 − 2 k is non-zero. This is true a s the term is t he pro duct of a collection of non- zero terms. It follo ws tha t the ab ov e equ ation implies ( d 2 − k e + de 2 − k ) 2 k +1 = 0, whic h gives d 2 − k e + de 2 − k = 0 for whic h the only solution is d = e . This con t radiction completes the pro of. 10 References [1] R. Bose and D. Ra y-Chaudh uri,“ O n a class of erro r correcting binary group co des”, Info. and Co ntr ol , 3 , pp 68- 79, 19 6 0. [2] C. Bra c ken, E. Byrne, N. Markin, G . McGuire, “Determining the Non- linearit y of a New F amily of APN F unctions,” Pr o c e e dings o f AAECC- 17 , Lecture Notes in Computer Science, V ol 4851 , pp 72 -79 , 2007. [3] H. D obb ertin, “Another pro of of Ka sami’s Theorem”, Designs , Co des and Crypto gr aphy , V ol. 17 , pp 177–180, 1999. [4] A. Ho cque nghem, “Co des correcteurs d’erreurs”, Chiffr es (Paris) , 2 , pp 147-156 , 1959. [5] V.S. Pless a nd W.C. Huffman, “Handb o ok o f Co ding Theory”, North Holland, Amsterdam, 1998. [6] R. 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