Abstract
This paper investigates the existence, enumeration and asymptotic performance of self-dual and LCD double circulant codes over Galois rings of characteristic $p^2$ and order $p^4$ with $p$ and odd prime. When $p \equiv 3 \pmod{4},$ we give an algorithm to construct a duality preserving bijective Gray map from such a Galois ring to $\mathbb{Z}_{p^2}^2.$ Using random coding, we obtain families of asymptotically good self-dual and LCD codes over $\mathbb{Z}_{p^2},$ for the metric induced by the standard $\mathbb{F}_p$-valued Gray maps.
Highlights
Double circulant self-dual codes over finite fields have been studied recently in [1]
We study these codes in the case of Galois rings of characteristic p2 and size p4, for p an odd prime
A recent topic related to self-dual codes is LCD codes
Summary
Double circulant self-dual codes over finite fields have been studied recently in [1]. Double circulant self-dual codes over a commutative ring can only exist if there is a square root of −1 over that ring [10] Such a root does not exist over Galois rings of even characteristic which are not fields, but does exist over Galois rings of odd characteristic and even extension degree [8, Lemma 3.1, Lemma 3.2]. We study LCD double circulant codes over the same Galois rings For every such Galois ring we construct a duality preserving Gray map with image Z2p2, which maps self-dual When n varies in one of these two families of primes, we obtain two infinite families of double circulant codes, one self-dual, one LCD, of length 2n.
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