We give a classification of four-circulant singly even self-dual [60, 30, d] codes for d = 10 and 12. These codes are used to construct extremal singly even self-dual [60, 30, 12] codes with weight enumerator for which no extremal singly even self-dual code was previously known to exist. From extremal singly even self-dual [60, 30, 12] codes, we also construct extremal singly even self-dual [58...

In this paper, we study ternary optimal formally self-dual codes. Bounds for the highest minimum weight are given for length up to 30 and examples of optimal formally self-dual codes are constructed. For some lengths, we have found formally self-dual codes which have a higher minimum weight than any self-dual code. It is also shown that any optimal formally self-dual [ 10,5,5] code is related t...

It is a well known fact that if C is an [n, k, d] formally self-dual even code with n > 30, then d ≤ 2[n/8]. A formally self-dual even code with d = 2[n/8] is called nearextremal. Kim and Pless [9] conjecture that there does not exist a near-extremal formally self dual even (not Type II) code of length n ≥ 48 with 8|n. In this paper, we prove that if n ≥ 72 and 8|n, then there is no near-extrem...

In [4], starting from an automorphism θ of a finite field Fq and a skew polynomial ring R = Fq[X; θ], module θ-codes are defined as left R-submodules of R/Rf where f ∈ R. In [4] it is conjectured that an Euclidean self-dual module θ-code is a θ-constacyclic code and a proof is given in the special case when the order of θ divides the length of the code. In this paper we prove that this conjectu...

The largest minimum weight of a self-dual doubly-even binary (n, k, d) code is d = 4bn/24c+ 4. Of such codes with length divisible by 24, the Golay Code is the only (24, 12, 8) code, the Extended Quadratic Residue Code is the only known (48, 24, 12) code, and there is no known (72, 36, 16) code. One may partition the search for a (48, 24, 12) self-dual doubly-even code into 3 cases. A previous ...

In this paper, we study the p-ary linear code C(PG(n, q)), q = p, p prime, h ≥ 1, generated by the incidence matrix of points and hyperplanes of a Desarguesian projective space PG(n, q), and its dual code. We link the codewords of small weight of this code to blocking sets with respect to lines in PG(n, q) and we exclude all possible codewords arising from small linear blocking sets. We also lo...

Dual codes are defined with respect to non-degenerate sesquilinear or bilinear forms over a finite Frobenius ring. These dual codes have the properties one expects from a dual code: they satisfy a double-dual property, they have cardinality complementary to that of the primal code, and they satisfy the MacWilliams identities for the Hamming weight. 2010 MSC: 94B05, 15A63

We show that every self-orthogonal code over $\mathbb F_q$ of length $n$ can be extended to a self-dual code, if there exists self-dual codes of length $n$. Using a family of Galois towers of algebraic function fields we show that over any nonprime field $\mathbb F_q$, with $q\geq 64$, except possibly $q=125$, there are self-dual codes better than the asymptotic Gilbert--Varshamov bound.

It is shown that the residue code of a self-dual Z4-code of length 24k (resp. 24k + 8) and minimum Lee weight 8k + 4 or 8k + 2 (resp. 8k + 8 or 8k + 6) is a binary extremal doubly even self-dual code for every positive integer k. A number of new self-dual Z4-codes of length 24 and minimum Lee weight 10 are constructed using the above characterization.

A classification of extremal double circulant self-dual codes of lengths up to 88 is known. We demonstrate that there is no extremal double circulant self-dual code of length 90. We give a classification of double circulant self-dual [90, 45, 14] codes. In addition, we demonstrate that every double circulant self-dual [90, 45, 14] code has no extremal selfdual neighbor of length 90. Finally, we...