Uncorrectable Errors of Weight Half the Minimum Distance for Binary Linear Codes

Uncorrectabl e Errors of W eight Half the Minimum Distance for Binary Linear Codes Kenji Y asunag a Graduate School of Science and T echnolo gy Kwansei Gakuin Univ ersity 2-1 Gakuen, Sanda, 669-13 37 Japan E-mail: yasunaga@kwansei.ac. jp T oru Fu jiwara Graduate School of Inform ation Science and T echn ology Osaka University 1-5 Y amadaok a, Suita, 56 5-0871 Japa n E-mail: fujiwara@ist.osaka-u.ac. jp Abstract — A lower bound o n the number of uncorr ectable errors of weight half the minimum distance is derive d for binary linear codes satisfying some c ondition. The condition is satisfied by some primitiv e BCH codes, extended primitive BCH codes, Reed-Mu ller codes, and random li near codes. Th e bound asymptotically c oincides with the corresponding upper bou nd fo r Reed-Muller codes and ra ndom linear codes. By generalizing the idea o f the l ower bound, a lo wer bound on th e number of uncorrectable error s for weights lar ger than h alf the m inimum distance is also obtained, but the generalized lower bound is weak for l arge weights. Th e monotone error structure and its related notion larger half and tri al set , which are in troduced b y Helleseth, Kløve, and Leve nshtein, are mainly used to derive the bounds. I . I N T R O D U C T I O N For a binary linear code, cor rectable er rors we consider here are bin ary errors correctable by the minimu m d istance decodin g, which perfor ms a maximum likelihoo d deco ding for bina ry symmetric channels. Syndr ome decodin g is on e of the min imum d istance decodin g. In syndrom e decod ing, the correctable errors ar e co set leader s of the code. When there ar e two o r more minimum we ight vectors in a coset, we have cho ices of the coset leader . If the lexicog raphically smallest minimu m weight vector is taken as the co set leader, then both the correctable errors and the un correctab le er rors have a m onoton e structure . Th at is, when y covers x (the support of y co ntains that o f x ), if y is correctab le, then x is a lso co rrectable, and if x is un correctab le, th en y is also uncor rectable [7] . Using this mono tone structu re, Z ´ emor showed th at the residual error pro bability af ter maximum likelihood decod ing displays a threshold behavior [ 12]. When uncorr ectable ( and co rrectable) er rors have the mon otone structure, th ey are characte rized by the m inimal unco rrectable (and maxima l corr ectable) errors. Larger h alves o f cod ew ords are in troduced by Helleseth , Kløve, and Levenshtein [5] to de- scribe th e minimal un correctab le erro rs. Th ey also introdu ced a tr ial set f or a code. It is a set of cod ew ords whose larger halves con tain all minimal u ncorrec table error s. Trial sets can be used f or a max imum likelihood d ecoding and f or giving a n upper bound on the num ber of uncorrectable errors. The set of a ll co dew ords excep t for the all-zero codeword and the set of min imal codew ords [1] in the code are examples of trial sets. In this paper, we stu dy bounds on the num ber of co r- rectable/un correctable erro rs. There were several works abo ut them. For the first-order Reed-Muller codes, the e xact numbers of correctab le errors of weight half the minimum distanc e and half the minimum distance plus one were determined [10], [11]. For gener al linear codes, some u pper bo unds o n the number o f uncor rectable errors were presen ted in [5], [4], [8 ]. In this work, we con sider lower b ounds on the nu mber of uncorr ectable errors based on the idea of [5] fo r general linear codes. W e deriv e a lower bound on the numb er of uncorr ectable errors o f weigh t half the minimum distan ce fo r cod es sat- isfying some co ndition. The bound is g i ven in terms of th e number s of codewords with weights d and d + 1 in a trial set fo r odd d , where d is the minimum distance of the code. For the case of e ven d , the bound is gi ven by the number of codewords with weight d in a trial set. Since the set of all codewords except the a ll-zero vector is a tr ial set, the bou nd can b e evaluated by the num bers of codew ords with weights d and d + 1 . The conditio n is not too r estricti ve, and some primitive BCH codes, extended primitive BCH codes, Reed- Muller co des, and rando m linear codes satisfy the condition. For Reed-Muller co des and random linear codes, the lower bound asymp totically coincid es with the u pper bo und of [5, Corollary 7]. The lower bound can be gen eralized to a lower bound on the size of the set of larger halves of a trial s et, which is a lower bou nd o n the n umber of uncorre ctable error s. In the next sectio n, we re view some definition s an d prop- erties of the mo notone error stru cture, larger halves, and trial sets. In Sec tion III, a lower boun d o n the numb er of uncorr ectable er rors of weigh t half the minimum distance is giv en for the codes satisfying some condition . The b ound presented in Section III is generalized in Section IV. I I . L A R G E R H A LV E S A N D T R I A L S E T S W e in troduce definitions and pr operties of larger halves and trial sets. Let F n be the set of all binary vectors of leng th n . Let C ⊆ F n be a binary linear code of length n , d imension k , an d m inimum d istance d . Then F n is partition ed into 2 n − k cosets C 1 , C 2 , . . . , C 2 n − k ; F n = S 2 n − k i =1 C i and C i ∩ C j = ∅ for i 6 = j , wh ere eac h C i = { v i + c : c ∈ C } with v i ∈ F n . The vector v i is called a co set leader of the coset C i if the weight of v i is smallest in C i . Let H be a parity check matr ix o f C . Th e syndr ome of a vector v ∈ F n is defin ed as v H T . All vector s ha ving the same syndrom e a re in the same coset. Syndrom e decod ing associates an erro r vector to each synd rome. The syndrom e decoder presumes that th e er ror vector added to the receiv ed vector y is th e coset lead er o f the coset which con tains y . Th e syndrom e decodin g fu nction D : F n → C is defin ed as D ( y ) = y + v i , if y ∈ C i . In this paper, we take as v i the minimum elemen t in C i with respect to the following total or dering  : x  y if and only if  w ( x ) < w ( y ) , or w ( x ) = w ( y ) an d v ( x ) ≤ v ( y ) , where w ( x ) denotes the Hamming weight of a vector x = ( x 1 , x 2 , . . . , x n ) and v ( x ) deno tes the nu merical v alue of x : v ( x ) = n X i =1 x i 2 n − i . W e write x ≺ y if x  y and x 6 = y . Let E 0 ( C ) b e the set of all coset leade rs of C . In th e syndrom e decoding, E 0 ( C ) is the set of c orrectable e rrors and E 1 ( C ) = F n \ E 0 ( C ) is the set of un correctab le e rrors. Since we take the minimum element with respect to  in each coset as its coset lead er , b oth E 0 ( C ) and E 1 ( C ) have the following well-known mono tone structure (see [7, Theorem 3.1 1]). Let ⊆ denote a partial orderin g called “covering” su ch that x ⊆ y i f and only if S ( x ) ⊆ S ( y ) , where S ( v ) = { i : v i 6 = 0 } is the support of v = ( v 1 , v 2 , . . . , v n ) . Consider x and y with x ⊆ y . If y is a correctable error, then x is also correctab le. If x is unco rrectable, then y is also uncorr ectable. Using this structure , Z ´ e mor showed that the r esidual error probab ility af ter maximum likelihood decoding displays a threshold behavior [12]. Helleseth, Kløve, and Levenshtein [5] studied this structure an d introduced la r ger halve s and trial sets . Since th e set of uncorrectab le errors E 1 ( C ) has a monoto ne structure, E 1 ( C ) can be charac terized by minimal uncor- r ectable err ors in E 1 ( C ) . An uncorrec table er ror y ∈ E 1 ( C ) is minima l if there exists no x such tha t x ⊂ y in E 1 ( C ) . W e denote by M 1 ( C ) the set of all minimal u ncorrectab le errors in C . Larger halves of a codew ord c ∈ C are intro duced to characterize the m inimal u ncorrec table errors, an d are defin ed as m inimal vectors v with respect to covering such that v + c ≺ v . The fo llowing con dition is a n ecessary and sufficient con dition that v ∈ F n is a larger half of c ∈ C : v ⊆ c , (1) w ( c ) ≤ 2 w ( v ) ≤ w ( c ) + 2 , (2) l ( v ) ( = l ( c ) if 2 w ( v ) = w ( c ) , > l ( c ) if 2 w ( v ) = w ( c ) + 2 , (3) where l ( x ) = min S ( x ) , (4) that is, l ( x ) is the lef tmost no n-zero coordin ate in the vector x . The co ndition (3) is no t applied if w ( c ) i s odd. The pr oof of equ i valence be tween the definition and the above conditio n is fou nd in th e proof of [5, Theo rem 1 ]. Let LH ( c ) be the set of all larger halves of c ∈ C . F or a set U ⊆ C \ { 0 } , define LH ( U ) = [ c ∈ U LH ( c ) . When the weight of a code word c is odd, the weight of the vectors in LH ( c ) is ( w ( c ) + 1) / 2 . When the weigh t of c is e ven, L H ( c ) consists of vectors of weights w ( c ) / 2 and w ( c ) / 2 + 1 . F or convenience, let L H − ( c ) and LH + ( c ) den ote the sets of larger halves of c of weight w ( c ) / 2 an d w ( c ) / 2 + 1 , respectively . Then LH ( c ) = LH − ( c ) ∪ LH + ( c ) . Also let LH − ( U ) = S c ∈ U LH − ( c ) and LH + ( U ) = S c ∈ U LH + ( c ) for a subset U o f e ven-weight subco de. A trial set T for a cod e C is defined as follows: T ⊆ C \ { 0 } is a tr ial set for C if M 1 ( C ) ⊆ LH ( T ) . Since e very larger half is an uncorrec table err or , we have the relation M 1 ( C ) ⊆ LH ( T ) ⊆ E 1 ( C ) . (5) In th e r est of paper , for u , v ∈ F n , we write u ∩ v as the vector in F n whose supp ort is S( u ) ∩ S( v ) . For a set U ⊆ F n , define A i ( U ) = { v ∈ U : w ( v ) = i } . Also we defin e M 1 i ( C ) = A i ( M 1 ( C )) and LH i ( U ) = A i ( LH ( U )) for U ⊆ C \ { 0 } . I I I . A B O U N D O N T H E N U M B E R O F U N C O R R E C TA B L E E R RO R S O F W E I G H T H A L F T H E M I N I M U M D I S TA N C E In this section, we deriv e a lower bou nd on | E 1 ⌈ d/ 2 ⌉ ( C ) | . The bo und is given by the nu mber of codewords w ith weigh ts d and d + 1 in a trial set. Since C \ { 0 } is a trial set for C , th e lower bound ca n be evaluated by the nu mber of codewords of weights d and d + 1 in C . Since the we ight ⌈ d/ 2 ⌉ is the minimum weight in E 1 ( C ) , ev ery vector in E 1 ⌈ d/ 2 ⌉ ( C ) is n ot covered by o ther unco r- rectable errors, and thu s M 1 ⌈ d/ 2 ⌉ ( C ) = E 1 ⌈ d/ 2 ⌉ ( C ) . F rom ( 5), we have M 1 ⌈ d/ 2 ⌉ ( C ) = LH ⌈ d/ 2 ⌉ ( T ) = E 1 ⌈ d/ 2 ⌉ ( C ) , where T is a trial set for C . W e will give a lo wer bound on | E 1 ⌈ d/ 2 ⌉ ( C ) | by giving a lower bound on | LH ⌈ d/ 2 ⌉ ( T ) | . A. Odd Minimum W eight Case When d is odd, LH ⌈ d/ 2 ⌉ ( T ) = LH ( A d ( T )) ∪ LH − ( A d +1 ( T )) . The next lemma implies that the n um- ber of common larger halves among LH ( A d ( T )) an d LH − ( A d +1 ( T )) is small. Lemma 1: L et C be a linear cod e with o dd minimum distance d . For e very c 1 , c ′ 1 ∈ A d ( C ) and c 2 , c ′ 2 ∈ A d +1 ( C ) , it holds that | LH ( c 1 ) ∩ LH ( c ′ 1 ) | = 0 , | LH ( c 1 ) ∩ LH − ( c 2 ) | ≤ 1 , and | LH − ( c 2 ) ∩ LH − ( c ′ 2 ) | ≤ 1 . Pr oof: F or c , c ′ ∈ C \ { 0 } , every vector v ∈ LH ( c ) ∩ LH ( c ′ ) has the prop erty that v ⊆ c ∩ c ′ . Since ev ery vector in LH ( c 1 ) , LH ( c ′ 1 ) , LH − ( c 2 ) , LH − ( c ′ 2 ) has weight ( d + 1) / 2 , it is e nough to sh ow that w ( c 1 ∩ c ′ 1 ) < ( d + 1) / 2 , w ( c 1 ∩ c 2 ) ≤ ( d + 1) / 2 , w ( c 2 ∩ c ′ 2 ) ≤ ( d + 1) / 2 . W e can prove them by using w ( c ∩ c ′ ) = ( w ( c ) + w ( c ′ ) − w ( c + c ′ )) / 2 and w ( c + c ′ ) ≥ d . From the previous lemma, we give a lower bound on the number of uncorre ctable errors of weight h alf the min imum distance. T he cor respondin g up per boun d is giv en uncond i- tionally by [5, Corollary 7]. Theor em 1: Let C be a linear code with odd minimum distance d and T be a trial set fo r C . If  d d +1 2  > | A d ( T ) | + | A d +1 ( T ) | − 1 (6) holds, then  d d +1 2  ( | A d ( T ) | + | A d +1 ( T ) | ) − (2 | A d ( T ) | + | A d +1 ( T ) | − 1 ) | A d +1 ( T ) | ≤ | E 1 d +1 2 ( C ) | ≤  d d +1 2  ( | A d ( T ) | + | A d +1 ( T ) | ) . Pr oof: Fro m Lemm a 1, a cod e word c ∈ A d ( T ) has at most one commo n larger half for every c ′ ∈ A d +1 ( T ) and does not have common larger halves for any c ′ ∈ A d ( T ) \ { c } . Thus at least | LH ( c ) | − | A d +1 ( T ) | vectors in LH ( c ) d oes not have commo n larger halves. Also, a codew ord c ∈ A d +1 ( T ) has at most one common larger half f or every c ′ ∈ A d ( T ) ∪ { A d +1 ( T ) \ { c }} , at least | LH − ( c ) | − | A d ( T ) | − | A d +1 ( T ) | + 1 vectors in L H − ( c ) does not hav e c ommon larger halves. For e very c 1 ∈ A d ( T ) and c 2 ∈ A d +1 ( T ) , we have | LH ( c 1 ) | = | LH − ( c 2 ) | =  d ( d +1) / 2  . Therefore we have the lower bou nd (  d ( d +1) / 2  − | A d +1 ( T ) | ) | A d ( T ) | + (  d ( d +1) / 2  − | A d ( T ) | − | A d +1 ( T ) | + 1) | A d +1 ( T ) | ≤ | LH ( d +1) / 2 ( T ) | = | E 1 ( d +1) / 2 ( C ) | . Th e up per b ound is obtained from the inequal- ity | LH ( d +1) / 2 ( T ) | = | LH ( A d ( T )) ∪ LH − ( A d +1 ( T )) | ≤ | LH ( A d ( T )) | + | LH − ( A d +1 ( T )) | ≤  d ( d +1) / 2  | A d ( T ) | +  d ( d +1) / 2  | A d +1 ( T ) | . The d ifference between the up per and lower boun ds is (2 | A d ( C ) | + | A d +1 ( C ) | − 1) | A d +1 ( C ) | . If the fractio n | A d +1 ( C ) | /  d ( d +1) / 2  tends to zero as the code leng th becomes large, the lo wer bou nd as ymptotically coin cides with the upper one. B. Even Minimum W eight Case When d is even, LH ⌈ d/ 2 ⌉ ( T ) = LH − ( A d ( T )) . The next lemma implies that th e number o f common larger halves among LH − ( A d ( T )) is small. Lemma 2: L et C be a linea r cod e with even minimum dis- tance d . For every c 1 , c 2 ∈ A d ( C ) , it holds that | LH − ( c 1 ) ∩ LH − ( c 2 ) | ≤ 1 . Pr oof: For contr adiction, suppose tha t there exist two distinct vectors in LH − ( c 1 ) ∩ L H − ( c 2 ) . Then it holds that w ( c 1 ∩ c 2 ) ≥ d/ 2 + 1 , b ut this leads to th e con tradiction that w ( c 1 + c 2 ) = w ( c 1 ) + w ( c 2 ) − 2 w ( c 1 ∩ c 2 ) ≤ d − 1 . Theor em 2: Let C be a linear code with e ven m inimum distance d . If 1 2  d d 2  > | A d ( T ) | − 1 (7) holds, then 1 2  d d 2  | A d ( T ) | − ( | A d ( T ) | − 1) | A d ( T ) | ≤ | E 1 d 2 ( C ) | ≤ 1 2  d d 2  | A d ( T ) | . Pr oof: Fr om Lemma 2, a codeword c ∈ A d ( T ) has at most on e common larger half for every c ′ ∈ A d ( T ) \ { c } . Thus at least | LH − ( c ) | − | A d ( T ) | + 1 vectors in LH − ( c ) does n ot h av e com mon larger h alves. Thu s we have the lower bound (  d d/ 2  / 2 − | A d ( T ) | + 1 ) | A d ( T ) | ≤ | L H − ( A d ( T )) | = | E 1 d/ 2 ( C ) | . The upp er bound is obtained from the inequality | E 1 d/ 2 ( C ) | = | LH − ( A d ( C )) | ≤  d d/ 2  | A d ( C ) | / 2 . The difference between the upp er and lo wer bounds is upper bound ed by | A d ( C ) | 2 . If th e fractio n | A d ( C ) | /  d d/ 2  tends to zero as the code len gth becom es large, the lower boun d asymptotically coincides with the upper one. When we take C \ { 0 } as a tria l set T , th e condition for a lower bo und can be weaker and the lower bound can be improved. Theor em 3: Let C be a linear code with e ven m inimum distance d . If 1 2  d d 2  >  | A d ( C ) | − 1 2  holds, then 1 2  d d 2  | A d ( C ) | −  | A d ( C ) | − 1 2  | A d ( C ) | ≤ | E 1 d 2 ( C ) | . Pr oof: Fr om Lemma 2, a code word c ∈ A d ( C ) has at most one common larger half for c ′ ∈ A d ( C ) \ { c } . If c and c ′ have the com mon larger half v ∈ LH − ( c ) ∩ LH − ( c ′ ) , then v is represented as v = c ∩ c ′ and it ho lds that l ( c ) = l ( c ′ ) . Then the other codew ord c + c ′ ∈ A d ( C ) does not have common larger halves with c , sin ce l ( c + c ′ ) 6 = l ( c ) . T herefor e at least | LH − ( c ) | − ⌈ ( | A d ( C ) | − 1) / 2 ⌉ vectors in LH − ( c ) does n ot have common larger halves. Thus we h av e the lower bou nd (  d d/ 2  / 2 − ⌈ ( | A d ( C ) | − 1) / 2 ⌉ ) | A d ( C ) | . T ABLE I T H E r - T H O R D E R R E E D - M U L L E R C O D E O F L E N G T H 2 m S AT I S F Y I N G ( 7 ) . r m 1 ≥ 4 2 ≥ 6 3 ≥ 8 4 ≥ 10 5 ≥ 11 6 ≥ 13 In what follows, we see that some BCH cod es, Reed-Muller codes, an d rand om linear c odes satisfy th e co nditions (6) or (7). For an ( n, k ) linear code C , which has code length n and dimension k , we choose C \ { 0 } as a trial set fo r C . 1) Primitive BCH cod es: By using the weig ht distribu- tion [2 ], we can verify that the ( n, k ) p rimitive BCH cod es satisfy the con dition (6) for n = 127 , k ≤ 64 and n = 63 , k ≤ 24 . 2) Extended Primitive BCH co des: By using the weig ht distribution [2], we can verify that th e ( n, k ) extended p rimi- ti ve BCH codes satisfy the condition (7) for n = 12 8 , k ≤ 6 4 and n = 64 , k ≤ 2 4 . 3) Reed-Muller codes: For the r -th o rder Reed-Muller code of len gth 2 m , the minimum d istance is 2 m − r and the num ber of minim um weigh t codewords | A 2 m − r (RM m,r ) | is presen ted in Theorem 9 of [ 6, Chapter 1 3], which is upper bo unded by (2 m +1 − 2) r . Th en, f or a fixed r , the co ndition (7) is satisfied except fo r small m . T able I shows wh ich p arameters mee t th e condition (7). The fraction | A d ( C ) | /  d d/ 2  is upper bounded by | A d ( C ) |  d d/ 2  ≤ (2 m +1 − 2) r 2 2 m − r ≤ 2 ( m +1) r − 2 m − r . Thus for a fixed r the f raction tends to zero as m beco mes large. This mea ns the up per and lower bou nds in Th eorem 2 asymptotically coincide. 4) Random Linear Cod es: A random linea r code is a code whose generato r matrix h as equ iprobab le entries. That is, first we set a par ameter ( n, k ) , and then we choose a generato r matrix fro m all the 2 nk possible gen erator matrices with probab ility 2 − nk . I t is known that with high probability the minimum distanc e equals to nδ GV , whe re 1 − H ( δ GV ) = k /n and H ( x ) is the binary en tropy f unction of x [3], [9]. Also it is k nown that the weight d istribution eq uals the binomial dis- tribution. Then, | A d ( C ) | ≈ (2 k − 1)  n d  2 − n ≈  n nδ GV  2 k − n ≈ 2 n ( H ( δ GV )+ k/n − 1) ≈ 1 , where we use the a pprox imation  n nλ  ≈ 2 H ( λ ) , and | A d +1 ( C ) | ≈ (2 k − 1)  n d +1  2 − n ≈  n nδ GV  2 k − n ( n − d ) / ( d + 1) ≈ 2 n ( H ( δ GV )+ k/n − 1) ≈ 1 . Since  d d/ 2  ≈ p 2 /π d 2 d ≈ 2 nδ for even d and  d ( d +1) / 2  ≈ 1 / p 2 π ( d + 1 )2 d +1 ≈ 2 nδ for odd d , wh ere d = nδ , the con ditions (6) and (7) are satisfied. Since the fr actions | A d +1 ( C ) | /  d ( d +1) / 2  and | A d ( C ) | /  d d/ 2  tend to zero, the upper and lower bound s in T heorems 1 and 2 asymptotically coincide. Remarks Note that the con dition (7) for T = C \ { 0 } is a suf ficient condition under wh ich e very codewords with weig ht d is contained in ev ery tr ial set for C with even min imum distance d . Also, th e co ndition (6) for T = C \ { 0 } is a suf ficient condition und er which e very codewords with weights d and d + 1 is con tained in e very trial set for C with od d minimu m distance d . When the condition (7) holds for T = C \ { 0 } , a s described in the proof o f Theorem 2, for every c ∈ A d ( C ) there exists at least one larger h alf v ∈ LH ⌈ d/ 2 ⌉ ( T ) that has no comm on larger h alf with other c odewords in C . Since M 1 ⌈ d/ 2 ⌉ ( C ) = LH ⌈ d/ 2 ⌉ ( T ) , e very larger h alf of c ∈ A d ( C ) is a minimal unc orrectable er ror . Every tr ial set T must satisfy that M 1 ( C ) ⊆ LH ( T ) . Therefore, e very cod e word in A d ( C ) needs to be contained in every trial set for C in this case. By a similar argument, we can show that if the co ndition (6) for T = C \ { 0 } holds, then e very codeword in A d ( C ) ∪ A d +1 ( C ) need to be in e very trial set f or C . I V . A G E N E R A L I Z A T I O N O F T H E B O U N D By gen eralizing th e results in th e previous section , we give a lower bound on the size of LH i ( C \ { 0 } ) for each i . W e have the relatio n M 1 i ( C ) ⊆ LH i ( C \ { 0 } ) ⊆ E 1 i ( C ) . Thus the following lower bou nd is also a lower bound on the n umber of uncorr ectable erro rs, but the boun d is wea k when i is large. Theor em 4: Let C be a line ar code with min imum distance d an d T be a trial set for C . Define B i = | A 2 i − 2 ( T ) | + | A 2 i − 1 ( T ) | + | A 2 i ( T ) | . For an integer i with ⌈ d/ 2 ⌉ ≤ i ≤ ⌊ n/ 2 ⌋ , if  2 i − 3 i  > 3  2 i − ⌈ d 2 ⌉ i  B i holds, then  2 i − 3 i  − 3  2 i − ⌈ d 2 ⌉ i  B i  B i ≤ LH i ( T ) ≤  2 i − 3 i  | A 2 i − 2 ( T ) | +2  2 i − 1 i  ( | A 2 i − 1 ( T ) | + | A 2 i ( T ) | ) Pr oof: First we observe that LH i ( T ) = LH + ( A 2 i − 2 ( T )) ∪ LH ( A 2 i − 1 ( T )) ∪ LH − ( A 2 i ( T )) . W e consider the up per b ound o n the num ber o f common larger h alves in L H i ( T ) . Let c , c ′ be codew ords in A 2 i − 2 ( T ) ∪ A 2 i − 1 ( T ) ∪ A 2 i ( T ) . Then w ( c ∩ c ′ ) = ( w ( c ) + w ( c ′ ) − w ( c + c ′ )) / 2 ≤ (2 i + 2 i − d ) / 2 = 2 i − ⌈ d/ 2 ⌉ . Therefo re the number of common larger halves of weight i between c and c ′ is at most  2 i −⌈ d/ 2 ⌉ i  . For c ∈ A 2 i − 2 ( T ) ∪ A 2 i − 1 ( T ) ∪ A 2 i ( T ) , the size of larger halves of c with weig ht i is at least  2 i − 3 i  . Since  2 i − 3 i  > 3  2 i −⌈ d/ 2 ⌉ i  B i , there is at least  2 i − 3 i  − 3  2 i −⌈ d/ 2 ⌉ i  B i larger halves of c with weight i that have no co mmon larger halves. Thus the lo wer bound follows. The upper bound is ob tained from the ineq uality | LH i ( T ) | ≤ | LH + ( A 2 i − 2 ( T )) | + | L H ( A 2 i − 1 ( T )) | + | LH − ( A 2 i ( T )) | ≤  2 i − 3 i  | A 2 i − 2 ( T ) | +  2 i − 1 i  | A 2 i − 1 ( T ) | +  2 i − 1 i  | A 2 i ( T ) | . V . 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