Double Circulant Self-Dual and LCD Codes Over ℤp2

We study double circulant LCD codes over [Formula: see text] for all odd primes [Formula: see text] and self-dual double circulant codes over [Formula: see text] for primes [Formula: see text]. We derive exact enumeration formulae, and asymptotic lower bounds on the minimum distance of the [Formula: see text]-ary images of these codes by the classical Gray maps.

On Self-Dual Four Circulant Codes

Four circulant codes form a special class of [Formula: see text]-generator, index [Formula: see text], quasi-cyclic codes. Under some conditions on their generator matrix they can be shown to be self-dual. Artin primitive root conjecture shows the existence of an infinite subclass of these codes satisfying a modified Gilbert–Varshamov bound.

Biorthogonal wavelet codes with prescribed code distance

We propose a scheme of construction of 2-circulant codes with given code distance on the basis of biorthogonal filters with the property of perfect reconstruction over a finite filed of odd characteristic. The corresponding algorithm for constructing biorthogonal filters utilizes the Euclidean algorithm for finding the gcd of polynomials.

ON LINEAR COMPLEMENTARY DUAL FOUR CIRCULANT CODES

We study linear complementary dual four circulant codes of length $4n$ over $\mathbb{F}_{q}$ when $q$ is an odd prime power. When $q^{\unicode[STIX]{x1D6FF}}+1$ is divisible by $n$, we obtain an exact count of linear complementary dual four circulant codes of length $4n$ over $\mathbb{F}_{q}$. For certain values of $n$ and $q$ and assuming Artin’s conjecture for primitive roots, we show that the relative distance of these codes satisfies a modified Gilbert–Varshamov bound.