The self-orthogonality and divisibility are two important properties of linear codes. It is interesting to establish relationship between them. By the well-known Gleason-Pierce-Ward Theorem, all self-dual divisible co...
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The self-orthogonality and divisibility are two important properties of linear codes. It is interesting to establish relationship between them. By the well-known Gleason-Pierce-Ward Theorem, all self-dual divisible codes have been totally classified. However, the relationship between the self-orthogonality and divisibility of a q-ary linear codes is known only for q = 2, 3 by Huffman and Pless in 2003. It has remained open for more than 20 years to consider other cases. The purpose of this paper is to settle this open problem under certain conditions and construct new families of self-orthogonal codes. Let q be a power of an odd prime p. Firstly, we prove that any pdivisible code containing the all-1 vector over the finite field F-q is self-orthogonal. More generally, it is concluded that any p-divisible [n, k] linear code over F-q containing codewords of weight n is monomially equivalent to an [n, k] self-orthogonal code over F-q. This result provides a very efficient way to find self-orthogonal codes from p-divisible codes. Secondly, we apply this result to construct self-orthogonal codes with excellent parameters or nice applications. For one thing, we use this result to study the self-orthogonality of generalized Reed-Muller codes, certain projective two-weight codes, and Griesmer codes. For another thing, by this useful result as well as the extending and augmentation techniques for linear codes, we construct eight new families of self-orthogonal divisible codes. These self-orthogonal codes and their duals contain many optimal or almost optimal codes. Besides, some self-orthogonal codes support combinatorial designs and some of them are proved to be optimal or almost optimal locally recoverable codes.
We classify all q-ary ?-divisible linear codes which are spanned by codewords of weight ?. The basic building blocks are the simplex codes, and for q = 2 additionally the first order Reed-Muller codes and the parity c...
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We classify all q-ary ?-divisible linear codes which are spanned by codewords of weight ?. The basic building blocks are the simplex codes, and for q = 2 additionally the first order Reed-Muller codes and the parity check codes. This generalizes a result of Pless and Sloane, where the binary self-orthogonal codes spanned by codewords of weight 4 have been classified, which is the case q = 2 and ? = 4 of our classification. As an application, we give an alternative proof of a theorem of Liu on binary ?-divisible codes of length 4? in the projective case.
It is shown that there does not exist a projective triply-even binary code of length 59. This settles the last open length for projective triply-even binary codes, which therefore exist precisely for the lengths 15, 1...
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It is shown that there does not exist a projective triply-even binary code of length 59. This settles the last open length for projective triply-even binary codes, which therefore exist precisely for the lengths 15, 16, 30, 31, 32, 45-51, and >= 60.
We construct some new linear codes over the field of order 7 to determine the exact value of the minimum length for which a linear code of dimension four with given minimum weight exists for some open cases. Most of t...
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ISBN:
(纸本)9781665402873
We construct some new linear codes over the field of order 7 to determine the exact value of the minimum length for which a linear code of dimension four with given minimum weight exists for some open cases. Most of the new codes are constructed as projective duals of some 7-divisible codes from some orbits of a projectivity in the projective space.
作者：
Chinen, KojiKindai Univ
Sch Sci & Engn Dept Math 3-4-1 Kowakae Higashiosaka Osaka 5778502 Japan
In this paper, first we formulate the notion of divisible formal weight enumerators and propose an algorithm for the efficient search of the formal weight enumerators divisible by two. The main tools are the binomial ...
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In this paper, first we formulate the notion of divisible formal weight enumerators and propose an algorithm for the efficient search of the formal weight enumerators divisible by two. The main tools are the binomial moments. It leads to the discovery of several new families of formal weight enumerators. Then, as a result, we find examples of extremal formal weight enumerators which do not satisfy the Riemann hypothesis. (C) 2019 Elsevier B.V. All rights reserved.
This paper establishes that there is no [98, 5, 72](4) code. Such a code would meet the Griesmer bound and the weights of its codewords would all be divisible by 4. The proof of nonexistence uses the uniqueness of cod...
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This paper establishes that there is no [98, 5, 72](4) code. Such a code would meet the Griesmer bound and the weights of its codewords would all be divisible by 4. The proof of nonexistence uses the uniqueness of codes with parameters [n, 4, n - 5](4), 14 less than or equal to n less than or equal to 17. The uniqueness of these codes for n greater than or equal to 15 had been established geometrically by others;but it is rederived here, along with that of the [14, 4, 9](4) code, by exploiting the Hermitian form obtained when the weight function is read modulo 2.
We establish an upper bound for the minimum distance of a divisible code in terms of its dual distance. The bound generalizes the Mallows-Sloane bounds for self-dual codes. We obtain a linear recurrence for the distan...
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We establish an upper bound for the minimum distance of a divisible code in terms of its dual distance. The bound generalizes the Mallows-Sloane bounds for self-dual codes. We obtain a linear recurrence for the distance distribution components of codes that attain the bound. From this we derive known conditions for the existence of extremal self-dual codes in a much simpler way. In the second half of the paper, we determine zeta functions for the codes that attain our new bound. Zeta functions for linear codes are defined in Duursma (Trans. Amer. Math. Soc. 351(9) (1999) 3609). Using properties of ultraspherical polynomials, we show that the zeta function of a quaternary extremal self-dual code has its zeros on the circle \T\=q(-1/2) in analogy with the zeta function of an algebraic curve. (C) 2002 Elsevier Science B.V. All rights reserved.
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