Weight enumerators of self-dual codes

The technique of using shadow codes to build larger self-dual codes is extended to codes over arbitrary fields. It is shown how to build the codes and how to determine the new weight enumerator as well. For codes over fields equipped with a square root of -1 and not of characteristic 2, a self-dual code of length n+2 can be built from a self-dual code of length n; for codes over a field without a square root of -1 and not of characteristic 2 a self-dual code of length n+4 is built from a self-dual code of length n; and for codes over fields of characteristic 2 the length of the new self-dual code depends on the presence of the all-one vector in the subcode chosen. In certain cases using the subcode of vectors orthogonal to the all-one vector, the new weight enumerator can be calculated directly from the original weight enumerator. Specific examples of the technique are illustrated for codes over F/sub 3/, F/sub 4/, and F/sub 5/. >

Gleason has recently shown that the weight enumerators of binary and ternary self-dual codes are polynomials in two given polynomials. In this paper it is shown that classical invariant theory permits a straightforward and systematic proof of Gleason's theorems and their generalizations. The joint weight enumerator of two codes (analogous to the joint density function of two random variables) is defined and shown to satisfy a MacWilliams theorem. Invariant theory is then applied to generalize Gleason's theorem to the complete weight enumerator of self-dual codes over GF(3) , the Lee metric enumerator over GF(5) (given by Klein in 1884!) and over GF(7) (given by Maschke in 1893!), the Hamming enumerator over GF(q) , and over GF(4) with all weights divisible by 2, the joint enumerator of two self-dual codes over GF(2) , and a number of other results.

In this paper, we use the graded ring construction to lift the extended binary Hamming code of length 8 to $$R_k$$ R k . Using this method we construct self-dual codes over $$R_3$$ R 3 of length 8 whose Gray images are self-dual binary codes of length 64. In this way, we obtain twenty six non-equivalent extremal binary Type I self-dual codes of length 64, ten of which have weight enumerators that were not previously known to exist. The new codes that we found have $$\beta = 1, 5, 13, 17, 21, 25, 29, 33, 41$$ β = 1 , 5 , 13 , 17 , 21 , 25 , 29 , 33 , 41 and 52 in $$W_{64,2}$$ W 64 , 2 and they all have automorphism groups of size 8.

Parametrization of self-dual codes by orthogonal matrices

The existence of optimal binary self-dual codes is a long-standing research problem. In this paper, we present some results concerning the decomposition of binary self-dual codes with a dihedral automorphism group $D_{2p}$, where $p$ is a prime. These results are applied to construct new self-dual codes with length $78$ or $116$. We obtain $16$ inequivalent self-dual $[78,39,14]$ codes, four of which have new weight enumerators. We also show that there are at least $141$ inequivalent self-dual $[116,58,18]$ codes, most of which are new up to equivalence. Meanwhile, we give some restrictions on the weight enumerators of singly even self-dual codes. We use these restrictions to exclude some possible weight enumerators of self-dual codes with lengths $74$, $76$, $82$, $98$ and $100$.

Dougherty, Gaborit, Harada, Munemasa and Sole (see ibid., vol.45, p.2345-60, 1999) have previously given an upper bound on the minimum Lee weight of a type IV self-dual Z/sub 4/-code, using a similar bound for the minimum distance of binary doubly even self-dual codes. We improve their bound, finding that the minimum Lee weight of a type IV self-dual Z/sub 4/-code of length n is at most 4[n/12], except when n=4, and n=8 when the bound is 4, and n=16 when the bound is 8. We prove that the extremal binary doubly even self-dual codes of length n/spl ges/24, n/spl ne/32 are not Z/sub 4/-linear. We classify type IV-I codes of length 16. We prove that all type IV codes of length 24 have minimum Lee weight 4 and minimum Hamming weight 2, and the Euclidean-optimal type IV-I codes of this length have minimum Euclidean weight 8.

Mass formulae for Euclidean self-orthogonal and self-dual codes over finite commutative chain rings

In this paper, we give some new extremal ternary self-dual codes which are constructed by skew-Hadamard matrices. This has been achieved with the aid of a recently presented modification of a known construction method. In addition, we survey the known results for self-dual codes over GF (5) constructed via combinatorial designs, i.e. Hadamard and skew-Hadamard matrices, and we give a new self-dual code of length 72 and dimension 36 whose minimum weight is 16 over GF (5) for the first time. Furthermore, we give some properties of the generated self-dual codes interpreted in terms of algebraic coding theory, such as the orders of their automorphism groups and the corresponding weight enumerators.

It is known that it is possible to construct a generator matrix for a self-dual code of length 2n+2 from a generator matrix of a self-dual code of length 2n. With the aid of a computer, we construct new extremal Type I codes of lengths 40, 42, and 44 from extremal self-dual codes of lengths 38, 40, and 42 respectively. Among them are seven extremal Type I codes of length 44 whose weight enumerator is 1+224y8+872y10+·. A Type I code of length 44 with this weight enumerator was not known to exist previously.

Codes over [formula omitted], Jacobi forms and Hilbert–Siegel modular forms over [formula omitted

Various constructions for self-dual codes over rings and new binary self-dual codes

Shadow Codes over [formula omitted

Euclidean and Hermitian self-dual MDS codes over large finite fields

Recently there has been interest in self-dual codes over finite rings. In this note, g-fold joint weight enumerators of codes over the ring ℤ k of integers modulo k are introduced as a generalization of the biweight enumerators. We establish the MacWilliams relations for these weight enumerators and investigate the biweight enumerators of self-dual codes over ℤ k . We derive Gleason-type theorems for the corresponding biweight enumerators with the help of invariant theory.

Construction of isodual codes over GF(q)