27,540 research outputs found

    Construction of quasi-cyclic self-dual codes

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    There is a one-to-one correspondence between ℓ\ell-quasi-cyclic codes over a finite field Fq\mathbb F_q and linear codes over a ring R=Fq[Y]/(Ym−1)R = \mathbb F_q[Y]/(Y^m-1). Using this correspondence, we prove that every ℓ\ell-quasi-cyclic self-dual code of length mℓm\ell over a finite field Fq\mathbb F_q can be obtained by the {\it building-up} construction, provided that char (Fq)=2(\mathbb F_q)=2 or q≡1(mod4)q \equiv 1 \pmod 4, mm is a prime pp, and qq is a primitive element of Fp\mathbb F_p. We determine possible weight enumerators of a binary ℓ\ell-quasi-cyclic self-dual code of length pℓp\ell (with pp a prime) in terms of divisibility by pp. We improve the result of [3] by constructing new binary cubic (i.e., ℓ\ell-quasi-cyclic codes of length 3ℓ3\ell) optimal self-dual codes of lengths 30,36,42,4830, 36, 42, 48 (Type I), 54 and 66. We also find quasi-cyclic optimal self-dual codes of lengths 40, 50, and 60. When m=5m=5, we obtain a new 8-quasi-cyclic self-dual [40,20,12][40, 20, 12] code over F3\mathbb F_3 and a new 6-quasi-cyclic self-dual [30,15,10][30, 15, 10] code over F4\mathbb F_4. When m=7m=7, we find a new 4-quasi-cyclic self-dual [28,14,9][28, 14, 9] code over F4\mathbb F_4 and a new 6-quasi-cyclic self-dual [42,21,12][42,21,12] code over F4\mathbb F_4.Comment: 25 pages, 2 tables; Finite Fields and Their Applications, 201

    Self-Dual Codes

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    Self-dual codes are important because many of the best codes known are of this type and they have a rich mathematical theory. Topics covered in this survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems, bounds, mass formulae, enumeration, extremal codes, open problems. There is a comprehensive bibliography.Comment: 136 page

    New extremal singly even self-dual codes of lengths 6464 and 6666

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    For lengths 6464 and 6666, we construct extremal singly even self-dual codes with weight enumerators for which no extremal singly even self-dual codes were previously known to exist. We also construct new 4040 inequivalent extremal doubly even self-dual [64,32,12][64,32,12] codes with covering radius 1212 meeting the Delsarte bound.Comment: 13 pages. arXiv admin note: text overlap with arXiv:1706.0169

    A new class of codes for Boolean masking of cryptographic computations

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    We introduce a new class of rate one-half binary codes: {\bf complementary information set codes.} A binary linear code of length 2n2n and dimension nn is called a complementary information set code (CIS code for short) if it has two disjoint information sets. This class of codes contains self-dual codes as a subclass. It is connected to graph correlation immune Boolean functions of use in the security of hardware implementations of cryptographic primitives. Such codes permit to improve the cost of masking cryptographic algorithms against side channel attacks. In this paper we investigate this new class of codes: we give optimal or best known CIS codes of length <132.<132. We derive general constructions based on cyclic codes and on double circulant codes. We derive a Varshamov-Gilbert bound for long CIS codes, and show that they can all be classified in small lengths ≤12\le 12 by the building up construction. Some nonlinear permutations are constructed by using Z4\Z_4-codes, based on the notion of dual distance of an unrestricted code.Comment: 19 pages. IEEE Trans. on Information Theory, to appea

    On the Residue Codes of Extremal Type II Z4-Codes of Lengths 32 and 40

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    In this paper, we determine the dimensions of the residue codes of extremal Type II Z4-codes for lengths 32 and 40. We demonstrate that every binary doubly even self-dual code of length 32 can be realized as the residue code of some extremal Type II Z4-code. It is also shown that there is a unique extremal Type II Z4-code of length 32 whose residue code has the smallest dimension 6 up to equivalence. As a consequence, many new extremal Type II Z4-codes of lengths 32 and 40 are constructed.Comment: 19 page
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