NEW SELF-DUAL [54, 27, 10] CODES EXTENDED FROM [52, 26, 10] CODES 1. v (denoted by wt( v ) ) is the number

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1 NEW SELF-DUAL [54, 7, 10] CODES EXTENDED FROM [5, 6, 10] CODES 1 Nikolay I. Yakov ABSTRACT: Usig [5, 6, 10] biary self-dual codes, possesig a automorphism of order 3, we costruct ew [54, 7, 10] biary self-dual codes. We do this by applyig a techiue for extedig a [k, k, d] self-dual code to a [k+, k+1] self-dual code. Most of the costructed codes have ew values of the parameter i their weight umerators,. We costruct ew codes with 0, 1, for the first weight fuctio ad with, 3 for the secod. KEYWORDS: codes, automorphisms, self-dual codes, weight eumerator INTRODUCTION Let be the -dimesioal vector space over the field liear [ k, ] code C is a k-dimesioal subspace of codeword. The Hammig weight of a vector of elemets. A. A elemet of C is called a v (deoted by wt( v ) ) is the umber of its o-zero coordiates. The miimum weight d of a code C is the smallest weight amog its ozero codewords. A code with miimum weight d is called a [, k, d ] code. A geerator matrix of a code C is a matrix whose rows form a basis of C. We say that a geerator matrix is i stadard form if Ik A, where I k deotes the kk idetity matrix. The weight eumerator W( y ) of a code C is give by W ( y) i Ai y where Ai v C wt( v) i i0. Two biary codes are called euivalet if oe ca be obtaied from the other by a permutatio of coordiates. The permutatio S is a automorphism of C, if C ( C) ad the set of all automorphisms of C forms a group called the automorphism group of C, which is deoted by Aut( C ) i this paper. Let ( uv, ) for uv, be a ier product i. We study biary self- u ( u,, u ), dual codes ad whe the base field is we have for 1 ( 1,, ) the followig Euclidea ier product: u v uivi i0 v v v The dual code of a [ k, ] code C is defied as C u ( u, v) 0 for all v C. (, ). 1 This research is supported by Shume Uivesity uder Project RD /

2 The dual C is a liear [, k] code. If C C, C is termed self-orthogoal, ad if C C, C is self-dual. Every self-dual code C have dimesio k. The largest possible miimum weights of sigly eve self-dual codes of legths up to 7 are determied i [1]. It was also show [] that the miimum weight d of a biary self-dual code of legth is bouded by 4 4, if (mod 4); 4 d (1) 4 6, if (mod 4). 4 We call a self-dual code meetig this upper boud extremal. A self-dual code is called optimal iff it is ot extremal but has the largest miimum weight for its legth. The weight eumerators of the optimal self-dual codes of legths from 5 ad 54 are kow [3], [4]: [5, 6, 10]: There are two possible forms for the weight eumerator, W5, y 7980 y 4800 y, W y y where , 1 (44 16 ) ( ), 0 1, 11. Codes exist for W 5,1 ad for W 5, whe 0,,9,1 (see [5]). [54, 7, 10]: There are two possible forms for the weight eumerator 10 1 W 1 (351 8 ) y ( ) y, where 0 43, 10 1 W54, 1 (351 8 ) y ( ) y, where Codes exist for W whe 0 19, 6 ad for W54, whe [1..1], 4, 6, 7 (see [6]). CONSTRUCTION METHOD I this paper, we exted biary self-dual codes of legth 5 havig a automorphism of order 3 to self-dual codes of legth 54 usig a techiue due to Harada ad Kimura [7]. Theorem 1 [7] Let be a geerator matrix of a self-dual code C of legth, ad let x ( x1,, x, x 1,, x ) be a vector i such that ( xx, ) 1, where (,) deotes the Euclidea ier product. Let yi ( x, ri) for 1i, where r i is the i-th row vector of. The the followig matrix geerates a self-dual code C' of legth. y y 1 0 x1 xi x y 1 1 y

3 Corollary 1 [7] Let S be a subset of the set 1,,, such that S is odd if 0(mod 4) ad S is eve if (mod 4). Let I A be a geerator matrix i stadard form of a self-dual code C of legth. Suppose that xi 1if i S ad xi 0 if i S ad that yi xi 1 for 1i. The the followig matrix: 1 0 x1 x 1 1 y1 y1 () I A y y geerates a self-dual code C' of legth +. EXTENDIN LENTH 5 CODES TO LENTH 54 I [5] we costructed some ew [5, 6,10] self-dual codes with automorphism of types 3 (14,10) ad 3 (16, 4). We use costructio () ad a exhaustive search o all odd cardiality sets S. The base codes which we exted are the followig: Case I: 640 codes with 8 havig a automorphism of type 3 (14,10). We have foud 8 ieuivalet [54,7,10] codes with weight eumerators W ad W 54, for differet values of. The codes with W 5,1 for 0,1,, ad W 5, for, 3 are the first kow codes with these weight fuctios. We give the umber of codes for differet values of (for all codes with 0 ) i Table 1 (the ew codes are i bolds). Number A10 1 of codes A W54,i Table 1: [54, 7, 10] self-dual codes with 0 from case I Case II: 36 codes with 9 havig a automorphism of type 3 (16,4). We have foud 35 ieuivalet [54, 7,10] codes with weight eumerators W ad W 54, for differet values of. The codes with W 5,1 for 0,, ad

4 W 5, for, 3 are the first kow codes with these weight fuctios. We give the umber of codes for differet values of i Table. Number A10 1 of codes A W54,i Table : [54, 7, 10] self-dual codes with 0 from case II Case III: 1 code with 1 havig a automorphism of type 3 (16,4). We have foud 75 ieuivalet ew codes. The codes with W for 0, 1 are the first kow codes with these weight fuctios. We give the umber of codes for differet values of i Table 3. Number A10 1 of codes A W54,i Table 3: [54, 7, 10] self-dual codes with 0 from case III We coclude with examples of codes for each ew weight distributio obtaied. To T costruct a geerator matrix of a code with W 5,i for j use ( I7 i, j ) T (removig the last colum of i, j cosistig of zeroes), where the matrices 1, e36caf599907cb e1697a5fbed4a eda6c5befded c8bc300066a3a416cefec1583c, c0797a34dcf4dcee179ab0 7e47b33fa9a148cc59843d446f c0ee8aec6cac eee66 1,1 0a8ce573659cf0000df bacd7e969f a0b9830 0c57ed65a37b7e7bfa8e8ac0e0 096e303c964eb4ea0310ea733f, 37e04a6f99cb9e97f765abea fa33d069d5cfe9ad90f69119fe eac848cc4ca6460e4c8eaa68a

5 1,,3 09bfdcb7af536c9990e ea6457bedb650088fdae fef69e1fd7bed41b0ba e9f06bdf36cd389bb830 0e7bcd8b4d7959bf64e8400 cfdda7d69ddd3ec1a b7c000c965c94da330da73f,, 908f0cf81dd367d457a8dcfc, dac3157a9f49e9cbe80a91a9 89e6d51f63b9573f4bf9dc681c 7a4c967ec801faf34ac5e9600bd 9bc80c034efbe16ab1a988a071e 468e46ca66ae6eea6444ca 6c6668ecce06ee0ccca8ae0ca 6ea6457bedb650088fdae e9f06bdf36cd389bb830 cfdda7d69ddd3ec1a f0cf81dd367d457a8dcfc are i hexadecimal form. 89e6d51f63b9573f4bf9dc681c 9bc80c034efbe16ab1a988a071e 6c6668ecce06ee0ccca8ae0ca Propositio There exist optimal biary self-dual [54, 7, 10] codes havig a weight distributio W whe 0, 6 ad for W 54, whe 1 7, 5. Ope problem Costruct or prove the oexistece of optimal biary self-dual [54, 7, 10] codes havig a weight distributio W with 3, 4, 5 ad for W 54, whe 5. REFERENCES 1. Coway J.H. ad Sloae N.J.A., A ew upper boud o the miimal distace of self-dual codes. // IEEE Tras. Iform. Theory 36, pp , Rais E.M., Shadow bouds for self-dual-codes. // IEEE Tras. Iform. Theory, vol. 44, pp , Bouyuklieva St., Harada M., Muemasa A., Restrictios o the weight eumerators of biary self-dual codes of legth 4m. // Proceedigs of the Iteratioal Workshop OCRT, White Lagoo, Bulgaria, pp , Huffma W.C., Automorphisms of codes with applicatio to extremal doubly-eve codes of legth 48. // IEEE Tras. Iform. Theory, vol. 8, pp , N. Yakov, New optimal [5, 6, 10] self-dual codes. // to appear.

6 6. Yakov N., Russeva R., Biary self-dual codes of legths 5 to 60 with a automorphism of order 7 or 13. // to appear i IEEE Tras. Iform. Theory. 7. Harada M., Kimura H., O extremal self-dual codes. // Math. J. Okayama Uiv., vol. 37, pp. 1-14, ABOUT THE AUTOR assist. prof. Nikolay I. Yakov, PhD, Kostati Preslavski Uiversity of Shume, 140 Saedieie Str, 9700 Shume, Bulgaria, jakov_iki@yahoo.com