Algebraic Geometry Codes: General Theory

Abstract

This chapter describes some of the basic properties of geometric Goppa codes, including relations to other families of codes, bounds for the parameters, and sufficient conditions for efficient error correction. Special attention is given to recent results on two-point codes from Hermitian curves and to applications for secret sharing.

Syndrome decoding for Hermite codes with a Sugiyama-type algorithm

“…The statement of Lemma 4.1 implies that (Λ, R) is a solution of the key equation (3). Therefore Proposition 3.1 implies that R − ΛU ∈ L((µ + )P + Q − (q + 1)(b m + 1)P 0 ).…”

“…This result is also proved in [18,Prop. 14] (see also [3,Section 1.4.6]). Recall that t = deg(Q) is the number of errors.…”

“…The last inequality uses the assumption on t. We now apply [3,Lemma 1.52]. This lemma states that for any divisors A, B such that deg(B) < (A + B) and (B) = 0, we have that (A) = 0.…”

Quantum codes from a new construction of self-orthogonal algebraic geometry codes

“…. ., N, then it holds that [12,Lemma 1.38]). Notice that a differential ω with these conditions does always exist.…”

Generalized Weierstrass semigroups and codes

“…This relates the subjects from the preceding sections, yielding an arithmetic/algebraic-geometric viewpoint to the combinatorial objects from the Ąrst section. For further references concerning these topics, we refer the reader to [HKT08;Duu08;MO15;vv88;Ser12;HVP98].…”