List decoding of a class of affine variety codes

Consider a polynomial $F$ in $m$ variables and a finite point ensemble $S=S_1 \times ... \times S_m$. When given the leading monomial of $F$ with respect to a lexicographic ordering we derive improved information on the possible number of zeros of $F$ of multiplicity at least $r$ from $S$. We then use this information to design a list decoding algorithm for a large class of affine variety codes.Comment: 11 pages, 5 table

On Steane-Enlargement of Quantum Codes from Cartesian Product Point Sets

In this work, we study quantum error-correcting codes obtained by using Steane-enlargement. We apply this technique to certain codes defined from Cartesian products previously considered by Galindo et al. in [4]. We give bounds on the dimension increase obtained via enlargement, and additionally give an algorithm to compute the true increase. A number of examples of codes are provided, and their parameters are compared to relevant codes in the literature, which shows that the parameters of the enlarged codes are advantageous. Furthermore, comparison with the Gilbert-Varshamov bound for stabilizer quantum codes shows that several of the enlarged codes match or exceed the parameters promised by the bound.Comment: 12 page

Steane-Enlargement of Quantum Codes from the Hermitian Curve

In this paper, we study the construction of quantum codes by applying Steane-enlargement to codes from the Hermitian curve. We cover Steane-enlargement of both usual one-point Hermitian codes and of order bound improved Hermitian codes. In particular, the paper contains two constructions of quantum codes whose parameters are described by explicit formulae, and we show that these codes compare favourably to existing, comparable constructions in the literature.Comment: 11 page