On the Weight Distribution of the Extended Quadratic Residue Code of Prime 137

On the W eight Distribution of the Extended Quadratic Residue Code of Prime 137 C . Tjhai, M. T omlinson, M. Ambroze and M. Ahmed Fixed and Mobil e Communications Research University of Plymouth Plymouth, PL4 8AA, Uni ted Kingdom P ost-pr int of 7th International ITG Conference on Source and Channel Coding , Ulm, 14–16 J anuary 2008 Abstract The Hamming weight enumerator function of the formally self-dual even, binary extended quadrati c residue code of prime p = 8 m + 1 is given by Gleason’s theorem for singly-even code . Using this theorem, the Hamming weight dis tribution of the extended quadra ti c residu e is completely determined once the nu mber of codewords of Hamming weight j A j , for 0 ≤ j ≤ 2 m , are known. The smal lest prime for which the Hamming weight di stribution of the correspondin g extended quadratic residue code is unknown is 137 . It is shown in this paper that, for p = 137 A 2 m = A 34 may be obtai ned without the need of exhaustive codeword enumeration. After the remainder of A j required by Gleason’s theorem are computed a nd independently verified us ing their co ngru- ences , the Hamming weight d istributi ons of the bina ry a ugmented and extended quadrati c residue codes of prime 137 are derived. 1 Introduction The Hamming w e ight distribution of a linear e rror corre cting co de is of practical and theore tical interest. It provides a great d eal of information on the cod e capability in detecting er r ors and in corre cting errors or erasures. The co mplexity of computing the Hamming weight d istribution of a code is expon e ntial. In gene ral, the computation requires one to enumerate all codewo rds of the code ; or to enumerate all co deword s of the dual and apply the MacWill iams id entity . Since the birth of c o ding theory , various algebraic error correcting codes have been discovered. One cla ssic family of s uch code s is the family of qu ad ratic re sidue (Q R) code s, w h ich has rich mathematical structure and good error correc ting capability . D e spite having these advantages, the con- struction of its algebraic decod er is no n trivial. Due to the e xistence of rich mathematical structure, there are considerable restrictions on the w eight 1 structure of this family of codes and therefore it is not ne cessary to enumer - ate all code w ords or those o f the d ual in computing the Hamming weight distribution. In fact, by knowing a fraction o f the Hamming we ight dis tri- bution, the complete distribution can be obtained . Recently , this method has bee n used by Gaborit et al [ 1 ] to obtain the Hamming weight d istribu- tions of binary e xtended QR cod e s of primes 73 , 89 , 9 7 , 1 13 and 1 27 1 . In our previous work [ 2 , 3 ], we have evaluated the Hamming weight distribu- tions of t he ex te nded QR cod es of prime s 151 and 167 . The smallest prime for w h ich the Ha mm ing weig ht distribution of the corre sponding extend ed QR cod e is not known in 137 and in this paper , its Hamming weight distri- bution is evaluated. W e show that even smaller fraction of the Hamming weight distribution is suffi c ie nt to de rive the co mplete Hamming weight distribution. The remainder of this paper is organised as fo llows. Section 2 gives the de finition and notation that we use in this paper –including a brief recall of the binary Q R code s . Section 3 discusses t he modular cong ruence of the number of codeword s of a given Hamming w eight and the Hamming weigh t distribution of the extended QR co de of prime 137 is de rived in Sec tion 4 . 2 Definition and Notation Let F n 2 be a space of v ector of length n wh ose elemen ts take v alue over F 2 (binary field) . An [ n, k , d ] binary linear code C of le ngth n , dime nsion k and minimum Hamming distance d , is a k -dimensional subspace of F n 2 . Let x , y ∈ F n 2 , the scalar prod u ct o f these two vectors is defined as x · y = P n − 1 j =0 x j y j (mo d 2) . Given a cod e C , the dual co de is de fined as C ⊥ = { c ⊥ | c · c ⊥ = 0 for all c ∈ C and c ⊥ ∈ F n 2 } . The hull of a code C is defined as H ( C ) = C ∩ C ⊥ . The Hamming weight of a vector v ∈ F n 2 , denoted by w t H ( v ) , is the number of its no n zero coo r dinates and the minimum Hamming distance of C is simply the smallest Hammin g weight of all codeword s in C . Throug hout this p aper , w e deal exclusively with Hamming space and for c onvenie nce, the word “Hamming” shall be omitted. The weight enume rator function of C is given by A C ( z ) = n X j =0 A j z j (1) where z is an indeterminate and A j is the number of codewords of weight j . The distributi o n of A j for 0 ≤ j ≤ n is called the weight d istribution of a code. Given a vector v ∈ F n 2 of ev en weight, if w t H ( v ) ≡ 0 (mod 4) , it is termed doubly-even ; otherwise w t H ( v ) ≡ 2 (mod 4) an d it is termed singly- even. An ev en code is o ne which has co deword s of eve n weight on ly . A code C is called self-dual if C = C ⊥ . A self-dual code may be doubly-e ven if the weight o f all co deword s is divisible by 4 or singly- even if the r e are 1 The Hamming weight distribu tion of tha t of pr ime 151 is also g iven in [ 1 ], but we have shown that this result has been incorrec tly reported, refer to [ 2 ] for the detailed discussion. 2 codewo rds whose w eight is congru ent to 2 (mo d 4) . In addition to self-dual code, there also exists formally self-dual code. A code C is termed f ormally self-dual if C 6 = C ⊥ but A C ( z ) = A C ⊥ ( z ) . 2.1 Quadratic Residue Codes In this subsection, a brief summary of QR codes over F 2 is give n [ 4 ]. Binary QR co des are cyclic co des of p r ime length p wher e p ≡ ± 1 (mo d 8) . Let Q and N be sets of qu ad ratic residue and non quadratic residue modulo p respectively . G iven a prime p , there are four QR co des denoted by Q p , N p , Q p and N p . If α is a primitive p -root of unity , the gene rator polyno mial of the [ p, ( p + 1) / 2 , d − 1] augme nted QR co des Q p and N p contains roots whose expone nts are element of Q and N respective ly . The [ p, ( p − 1 ) / 2 , d ] expurgate d QR cod es Q p and N p contain, in their ge nerator polyn o mial, α 0 in addition to the roots of the respective au g mented QR cod es. No te that Q p (resp. Q p ) is permutation equivalent to N p (resp. N p ). If p ≡ − 1 (mod 8) , Q ⊥ p = Q p and as such the [ p + 1 , ( p + 1) / 2 , d ] extende d QR cod e ˆ Q p is self-dual and doubly-e ven. F or p ≡ 1 (mod 8 ) , Q ⊥ p = N p and there fore ˆ Q p 6 = ˆ Q ⊥ p but A ˆ Q p ( z ) = A ˆ Q ⊥ p ( z ) implying the co r re- sponding extende d QR code is formally self-dual. In this paper , we are interested in the QR co des where p ≡ 1 (mo d 8) , in particular p = 1 37 . Since the ex tended code is formally self-dual, the re- strictions o n the weight structure imposed by Gleason’s theorem fo r singly- even co de applies . This imp lies that for a g iven prime p = 8 m + 1 , the w eight enumerator function A ˆ Q p ( z ) is giv en by [ 5 ] A ˆ Q p ( z ) = m X j =0 K j (1 + z 2 ) 4 m − 4 j +1 { z 2 (1 − z 2 ) 2 } j (2) for some intege r K j . Equation ( 2 ) sho ws that the complete weight distribu- tion can be derived once the first m ev en terms of A j ( A 0 = 1 by definition) are known. Note that ˆ Q p is an even code and thus A j = 0 for odd integer j . 3 Congruence of the Number of Code words of a Given W e ight It is known in the literature that the automorphism gro up of ˆ Q p , denoted by Aut( ˆ Q p ) , contains the projective spec ial linear group PSL 2 ( p ) [ 4 ]. This lin- ear group is g enerated by a set of permutations on the coordinates ( ∞ , 0 , 1 , . . . , p − 1) of the form y → ( ay + b ) / ( cy + d ) where a, b, c, d ∈ F p , y ∈ F p ∪ { ∞ } an d ad − bc = 1 . This set of permutations may be produced by the transforma- tions 2 S : y → y + 1 and T : y → − y − 1 . The knowled ge of the automo rphism group of a code may be exp lo ited to characterise the weight distribution of the code. 2 In some cases, w e can see th at, in additi on to S and T , the transformation V : y → ρ 2 y where ρ is a g enerator of F p also g enerates the desired permutation of PSL 2 ( p ) . However , strictly speaking, V is redundant since V = T S ρ T S µ T S ρ where µ = ρ − 1 (mod p ) . 3 Let Aut( ˆ Q p ) ⊇ PSL 2 ( p ) = H , the number o f weig h t j codew ords A j can be categorised into two class e s: one which co ntains all we ight j cod e - words that are invariant u nder some ele ment of H and another which con - tains the rest. Given a codewo rd of ˆ Q p that is no t invariant u nder some element of H , apply in g all |H | = 1 2 p ( p 2 − 1) permutations will result in |H| distinct codewords o f ˆ Q p . In other wor d s , the latter cl ass fo rms orbits of size equal to the cardinality of PSL 2 ( p ) . Let A j ( H ) denote the n umber o f weight j codew ords wh ich are invariant under some element of H , we may write A j = n j · |H| + A j ( H ) ≡ A j ( H ) (mo d 1 2 p ( p 2 − 1 )) (3) for n j ∈ Z ∗ = { 0 } ∪ Z + i.e. non negative integer . Since |H | can be factorised as H = Q i q e i i where q i is a pr ime and e i is a po sitive integer , it is shown in [ 6 ] that A j ( H ) may be obtained by ap plying the Chin e se Re mainder The - orem to A j ( S q i ) (mo d q e i i ) for all primes q i that d ivide |H| . Note that S q i is the Sylow - q i -subgroup of H and A j ( S q i ) is the number o f code words of weight j fixed by some element of S q i . F or each prime q i , in order to co mpute A j ( S q i ) , the subcode which is invariant under s om e element of S q i needs to be obtained. F or odd primes q i , S q i is cyclic and there ex ists  a b c d  ∈ H , f or some inte gers a, b, c, d , which generates cyclic permutation of orde r q i . Thus, it is st raigh tfo rward to ob- tain the invariant subcode and the correspon ding A j ( S q i ) . On the other hand, if q i = 2 , S 2 is a dihedral group of order 2 s , where s is the highest power of 2 that divides |H| , and A j ( S 2 ) is given by [ 6 ] A j ( S 2 ) ≡ (2 s − 1 + 1 ) A j ( H 2 ) − 2 s − 2 A j ( G 0 4 ) − 2 s − 2 A j ( G 1 4 ) (mod 2 s ) , (4) where H 2 and G i 4 , for i = 0 , 1 , are subgro ups of order 2 and 4 respectively , which are contained in S 2 . Let P ∈ H of orde r 2 s − 1 and T =  0 − 1 1 0  ∈ H of order 2 , it is sho wn in [ 6 ] that H 2 = { 1 , P 2 s − 2 } and the non cyclic subgroup G i 4 = { 1 , P 2 s − 2 , P i T , P 2 s − 2 + i T } . 4 The W eight Distribution F ollowing G leason’s theorem, see ( 2 ), the weig ht distribution of the binary extende d QR code of prime 137 is given by A ˆ Q 137 ( z ) = 17 X j =0 K j (1 + z 2 ) 69 − 4 j ( z 2 − 2 z 4 + z 6 ) j . (5) Since A 0 = 1 and the minimum distance of ˆ Q 137 is 22 , only A 2 j , fo r 11 ≤ j ≤ 17 , are re quired in orde r to d educe A ˆ Q 137 ( z ) completely . Note that each A 2 j determines K j for some intege r j . Howev er , following the idea in [ 6 ] which has been relatively forg otten, K 17 may be determined without the need of exhaustively computing A 34 as shown in this sectio n . Let us first de duce the modular congrue nce of A 2 j , for 11 ≤ j ≤ 17 , of ˆ Q 137 . Some of these congruence s ha ve been given in the autho rs’ previous 4 work [ 2 ], but are restated in the f ollowing to make the paper self-co ntained. F or p = 137 , it is clear that |H | = 2 3 · 3 · 17 · 23 · 137 = 1285 608 . Let P =  0 37 37 31  and let  0 1 136 1  ,  0 1 136 6  and  0 1 136 11  be generators of permutation of orde rs 3 , 17 and 23 re spectively . It is not ne cessary to find a gen erator that generates permutation of or d er 137 as it fi x es the all zeros and all ones co deword s only . Subcodes that are invariant un der H 2 , G 0 4 , G 1 4 , S 3 , S 17 and S 23 are obtained and the number of weight 2 j , for 11 ≤ j ≤ 17 , c o dewor d s in these subcodes are the n c o mputed. The results are tabulated as fo llows, where k denotes the dimension of the corresponding subcode, H 2 G 0 4 G 1 4 S 3 S 17 S 23 S 137 k 35 19 18 23 5 3 1 A 22 170 6 6 0 0 0 0 A 24 612 10 18 46 0 0 0 A 26 1666 36 6 0 0 0 0 A 28 8194 36 60 0 0 0 0 A 30 34816 126 22 943 0 0 0 A 32 11456 3 261 1 89 0 0 0 0 A 34 34345 3 351 39 0 2 0 0 . F or p = 137 , ( 4 ) becomes A 2 j ( S 2 ) ≡ 5 A 2 j ( H 2 ) − 2 A 2 j ( G 0 4 ) − 2 A 2 j ( G 1 4 ) (mod 8 ) and using this formulation, the f o llowing congruence s A 22 ( S 2 ) = 2 (mod 8) A 24 ( S 2 ) = 4 (mod 8) A 26 ( S 2 ) = 6 (mod 8) A 28 ( S 2 ) = 2 (mod 8) A 30 ( S 2 ) = 0 (mod 8) A 32 ( S 2 ) = 3 (mod 8) A 34 ( S 2 ) = 5 (mod 8) are obtained. Combining all the abov e results using the Chinese-Remainder-Theorem, it follows that A 22 = n 22 · 1285608 + 3 21402 A 24 = n 24 · 1285608 + 1 07134 0 A 26 = n 26 · 1285608 + 9 64206 A 28 = n 28 · 1285608 + 3 21402 A 30 = n 30 · 1285608 + 4 28536 A 32 = n 32 · 1285608 + 1 12490 7 A 34 = n 34 · 1285608 + 1 14381 3 (6) for some non negative integers n 2 j . Let G be the g enerator matrix of the h alf-rate cod e ˆ Q 137 . In order to efficiently count the number of code words of weight 2 j , two full-rank ge ner- ator matrices, sa y G 1 and G 2 , w hich have p airwise disjoint information sets 5 are required. These matrices can be easily obtained by performin g Gaus- sian elimination o n G to p roduce G 1 = [ I | A ] and re peating the p rocess o n submatrix A to p roduce G 2 = [ B | I ] . F or e ach of these full-rank matrices, we need to enumerate as many as j X i =0  69 i  codewo rds and count the num be r of tho se of w eight 2 j . The e f ficiency of enumeration may be improved by employing the revolving door combina- tion generator algo rithm [ 7 ], which has the pro p erty that in two successive combination patterns , there is only one element that is exchanged. In ad- dition to this, the revolv ing door algo rithm also has a nice pr o perty that allows the enumeration to be realised on grid co mputer , s e e Appendix A.1 . W e have evaluated A 2 j , fo r 11 ≤ j ≤ 16 , using a grid o f appro ximately 1500 computers and the results are g iven below A 22 = 3214 02 A 24 = 2356 948 A 26 = 2153 3934 A 28 = 4901 38050 A 30 = 6648 30750 4 A 32 = 7786 52590 35 . (7) Comparing ( 6 ) and ( 7 ), it can be clearly seen that 3 n 22 = 0 , n 24 = 1 , n 26 = 16 , n 28 = 381 , n 30 = 5171 and n 32 = 6056 6 . The non negative integ er solutions of n 2 j give an ind ication that the cor r esponding A 2 j has been accurately computed. W e no w show that A 34 is kno wn. It is wo rth no ting that knowing A 34 , based o n the argumen ts on code word counting give n above, signifi- cantly reduce s the comp lexity of computing A ˆ Q 137 ( z ) . Co n sider Gleason’s formulation give n in ( 5 ), if we take its fir st deriv ative with r espect to z , we have d dz A ˆ Q 137 ( z ) = 17 X j =0 K j (1 + z 2 ) 68 − 4 j ( z 2 − 2 z 4 + z 6 ) j − 1 n 2(69 − 4 j ) z ( z 2 − 2 z 4 + z 6 )+ j (1 + z 2 )(2 z − 8 z 3 + 6 z 5 ) o (8) 3 Note that A 2 j , for 11 ≤ j ≤ 16 , h ave also been given in [ 1 ], however , A 30 and A 32 have been incorrectly reported as demonstrated in [ 2 ]. 6 which may be exp anded as d dz A ˆ Q 137 ( z ) = (1 + z 2 ) 68 K 0 + (1 + z 2 ) 64 n 130 z ( z 2 − 2 z 4 + z 6 )+ (1 + z 2 )(2 z − 8 z 3 + 6 z 5 ) o K 1 + (1 + z 2 ) 60 ( z 2 − 2 z 4 + z 6 ) n 122 z ( z 2 − 2 z 4 + z 6 )+ 2(1 + z 2 )(2 z − 8 z 3 + 6 z 5 ) o K 2 + . . . ( z 2 − 2 z 4 + z 6 ) 16 n 2 z ( z 2 − 2 z 4 + z 6 )+ 17(1 + z 2 )(2 z − 8 z 3 + 6 z 5 ) o K 17 . (9) From ( 9 ), we can see that the te rms that involve K j for 0 ≤ j ≤ 16 be come zero if we set z = i = √ − 1 . Thus, d dz A ˆ Q 137 ( z )    z = i = 2 i ( i 2 − 2 i 4 + i 6 ) 17 K 17 = − i 2 35 K 17 . (10) Since Aut( ˆ Q p ) is do ubly-transitive, given A 2 j of an extended QR c o de ˆ Q p , the numbe r of cod e words of weig ht 2 j − 1 and 2 j in the augmented code Q p are 2 j p +1 A 2 j and p +1 − 2 j p +1 A 2 j respectively . F o llowing [ 8 ], the weight enumerator function of Q 137 may be written in terms of that of ˆ Q 137 as follows A Q 137 ( z ) = A ˆ Q 137 ( z ) +  1 − z 138  d dz A ˆ Q 137 ( z ) . (11) From ( 5 ), it is obvio us that A ˆ Q 137 ( z )    z = i = 0 and therefore ( 11 ) be comes A Q 137 ( z )    z = i = − i 1 − i 138 2 35 K 17 . (12) The expur gated QR c o de Q 137 is an e ven code and fo llowing [ 4 ], Q ⊥ 137 = N 137 . W e can see that the expo nents o f the ze ros of Q 137 are in the set Q ∪ { 0 } , where as those of N 137 are in the set N , and thus the hull of Q 137 has dime nsion zero. It follows from [ 9 , Lemma 7.8.3 pp. 276] that the code Q 137 may be decomp osed into an orthogonal sum of eithe r 34 subcodes each consisting of three doubly-ev en and one singly-even code w ords; or 33 sub- codes each consisting o f three dou bly-even and one singly-e v en co deword s , in addition to o ne subco de c ontaining o ne doubly-ev en and thre e singly- even codeword s . As a c o nsequence, if W w denotes the numbe r o f codeword s of weight cong ruent to w (mo d 4) in Q 137 , we have, see [ 9 , Theore m 7.8.6 pp. 277] W 0 − W 2 = ± 2 34 . (13) 7 Note that this result also holds for Q 137 as Q 137 is the even weight subcode of Q 137 . Since all one s codeword 1 p ∈ Q 137 , it f ollows that W 1 − W 3 = ± 2 34 (14) for the augmented QR code. Substituting z w ith i in the weight enumerator function of Q 137 , we have A Q 137 ( z )    z = i = A 0 + i A 1 − A 2 − i A 3 + A 4 + i A 5 − A 6 − i A 7 + . . . − A 130 − i A 131 + A 132 + i A 133 − A 134 − i A 135 + A 136 + i A 137 = h X j ≡ 0 mo d 4 A j − X j ≡ 2 mo d 4 A j i + i h X j ≡ 1 mo d 4 A j − X j ≡ 3 mod 4 A j i = [ W 0 − W 2 ] + i [ W 1 − W 3 ] and thus, follow in g ( 13 ) and ( 14 ), A Q 137 ( z )    z = i = ± 2 34 (1 + i ) . (15) Equating ( 12 ) and ( 15 ), − i 1 − i 138 2 35 K 17 = ± 2 34 (1 + i ) , we arrive at K 17 = ∓ 69 . (16) Using ( 7 ), A 2 j = 0 for 1 ≤ j ≤ 10 and A 0 = 1 , K j for 0 ≤ j ≤ 16 are determined . Substituting these into ( 5 ) and equating the coefficients of z 34 with A 34 , we have A 34 = 7710 68968 296 + K 17 . (17) Consider the case for K 17 = − 69 , A 34 = 7710 68968 227 . Comparing this A 34 with the congrue nce g iven in ( 6 ), it fo llo ws that n 34 6∈ Z ∗ and h e nce this rules out the possibility of K 17 = − 69 . If K 17 = 69 , howeve r , A 34 = 7710 68968 365 (18) and it follows that n 34 = 5997 69 ∈ Z ∗ , indicating that K 17 is indeed 69 . Now we ha ve dete rmined A 34 (and hence K 17 ) without exhaustively counting the num ber of codewo rds of weight 34 in ˆ Q 137 . The weight d is- tribution of ˆ Q 137 can be straightfo rwardly deduced from ( 5 ) and so is that of Q 137 from ( 11 ). The weight distribut io ns of the augme nted and also the extende d QR code of prime 137 are tabulated in T able 1 . Note that since the we ig ht distributions are symmetrical, only the first half terms are tab - ulated. 8 Acknowledgements The authors wish to thank the PlymG RID team of the Unive rsity of Ply- mouth for providing the high p erformanc e computing resources. References [1] P . Gaborit, C.-S . Nedeloaia, and A. W assermann, “ O n the we ig ht enu- merators of duadic and quadratic residue cod e s ,” IEEE T rans . Inform. Theory , vo l. 51, pp. 402–407, J an. 2005. [2] C . Tjhai, M. T omlinson, R. Horan, M. Ahme d, and M. Ambroze, “Some results o n the weight distributions of the binary double- circulant co des based on pr imes, ” in Proc. 10 th IEEE Internation al Conference on Com - municatio ns S y stems , ( Singapore), 30 Oct.–1 Nov 2006. [3] C . Tjhai, M. T omlinson, R. Horan, M. Ahmed , and M. Ambroze, “On the e f ficient code words co unting algorithm and the weight d istribu- tion of the binary quad ratic double-c irc u lant co d es, ” in Proc . IEEE In- formation Theory W orkshop , (Che ngdu, China), pp. 42–46, 22–26 Oct. 2006. [4] F . J . MacWilliams and N . J . A. Slo an e, T he Theory o f Erro r -Correcting Codes . No rth-Holland, 1977. [5] E. M. Rains and N . J . A. Sloane, “Self-D u al Code s, ” in Han dbook of Coding Theory (V . S . Pless and W . C . Huff man, e ds.), Elsevier , North Holland, 1998. [6] J . Mykkeltveit, C . Lam, and R . J . McEliece, “On the weig ht enu- merators of quadratic re sidue codes, ” JPL T echnica l Repo rt 32-1526 , vol. XII, pp. 161–166, 1972. [7] A. N ijenhuis and H. S . W ilf, Combinatori al Algorithms for C omputers and Calculators . Academic Press , L o ndon, 2 nd ed., 1978. [8] J . H. van Lint, “Coding theor y , ” in Lecture Notes in Mathematics No . 201 , Springer , Be rlin, 1970. [9] W . C. Hu f fman and V . S . Pless, Fundamentals of Erro r -Correcting Codes . Cambridg e Univ e rsity Press, 2003. ISBN 0 521 78280 5. [10] H. Lüneburg, “Gray code s ,” Abh. Math. Sem. Hamburg , vol. 52, pp. 208–227, 1982. [11] D . E. K nuth, The Art o f Co mputer Programmin g , V ol. 4: F as cicle 3: Generating All Combinations and P artitio ns . Ad dison-W e sley , 3 rd ed., 2005. ISBN 0 201 85394 9. 9 A Appendix A.1 P arallel Realisa tion of Codeword Enumeration In this appendix, a method to e numerate code words in parallel is described and for a detailed description, refe r to [ 7 , 10 , 11 ]. Let C s t denote the combi- nation of t ou t of s elements with the com bination pattern represented by an orde red set a t a t − 1 . . . a 1 , where a 1 < a 2 < . . . < a t − 1 < a t . A pattern is said to have rank r if this pattern appears as the ( r + 1) th eleme nt in the list o f all C s t combinations. Here, it is assumed that the first element in the list of all C s t combinations has rank 0 . The combination C s t , wh ich f ollows the revolving door constraint and has an ordered set pattern, exhibits the following property C s t ⊃ C s − 1 t ⊃ . . . ⊃ C t +1 t ⊃ C t t . Consequently , this implies that, for the revolving door combination patterns of the for m a t a t − 1 . . . a 1 , if those o f fi x ed a t are consider ed, the maximum and minimum ranks o f such patterns are  a t +1 t  − 1 and  a t t  respectively . Let Rank( a t a t − 1 . . . a 1 ) be the rank of the pattern a t a t − 1 . . . a 1 , the re- volving doo r combination also has the following rec u rsive property on its rank, Rank( a t a t − 1 . . . a 1 ) =  a t + 1 t  − 1  − Rank( a t − 1 . . . a 1 ) . (19) As an implication of this , if all  k t  codewo rds need to be enu merated, for some integers k , t > 0 and k ≥ t , we can split the enumeration into ⌈  k t  / M ⌉ blocks where in e ach blo ck only at most M codew ords need to be en u mer - ated. In this way , the enumeration o f each block can be done o n a sep arate computer–allowing parallelism of codew ord enume ration. W e kno w that at the j th block, the enumeration w ould start from rank ( j − 1 ) M and the cor- responding pattern can be easily o btained by mak ing use of ( 19 ) as w ell as the maximum and minimum ranks o f the patterns of fi x ed a t . 10 T able 1: The weight distributions of [137 , 69 , 21] augme n ted and [138 , 69 , 22 ] extende d quadratic residue codes j Q 137 = [137 , 69 , 21] ˆ Q 137 = [138 , 69 , 22] 0 1 1 21 51238 0 22 27016 4 321402 23 40990 4 0 24 19470 44 2356948 25 40571 18 0 26 17476 816 21533934 27 99448 300 0 28 39068 9750 490138050 29 14452 8424 0 0 30 52030 2326 4 66483 0750 4 31 18055 7122 40 0 32 59809 5467 95 77865 2590 35 33 18997 3513 945 0 34 58109 5454 420 77106 8968 365 35 17092 08146 190 0 36 48427 56414 205 65519 6456 0395 37 13221 9821 02853 0 38 34794 6897 44350 48016 6718 47203 39 88328 7008 33460 0 40 21640 5317 041977 30473 4017 875437 41 51198 0845 799941 0 42 11702 4193 3257008 16822 2277 9056949 43 25853 7436 0137184 0 44 55232 9976 9383984 81086 7412 9521168 45 11414 8647 29214318 0 46 22829 7294 58428636 34244 5941 87642954 47 44202 3803 61406672 0 48 82879 4631 77637510 12708 1843 539044182 49 15053 5995 889831600 0 50 26494 3352 766103616 4154 7934 86559 35216 51 45196 1780 387038844 0 52 74747 5252 178564242 11 99437 0325 65603086 53 11987 8183 0242451728 0 54 18647 7173 5932702688 306 35535 6617 5154416 55 28141 1049 1202421488 0 56 41206 6179 0689260036 693 47722 8189 1681524 57 58556 7546 9990794812 0 58 80767 9375 1711441120 139 32469 22170 2235932 59 10814 6906 10004223000 0 60 14059 0977 93005489900 24 87378 84030 09712900 61 17746 7319 37729182608 0 62 21754 0585 04313191584 39 50079 04420 42374192 63 25897 6867 19588958304 0 64 29944 2002 69524733039 55 84188 69891 13691343 65 33629 6395 51783390742 0 66 36686 8795 11036426264 70 31651 90628 19817006 67 38877 1429 78140092004 0 68 40020 5883 59850094710 78 89773 13379 90186714 11