Decreasing norm-trace codes

TL;DR

This work studies decreasing norm-trace codes, a versatile family of evaluation codes defined by monomial sets closed under divisibility on the extended norm-trace curve ${\mathcal{X}}_u$. It employs Grobner basis theory and standard indicator functions to derive explicit parameters: length $n = q^{r-1}((q-1)u+1)$, dimension $k = |\mathcal{M}|$, and minimum distance $d = ((q-1)u+1)q^{r-1} - \max\{ \min(a q^{r-1} + (u(q-1) + 1 - a) b, a q^{r-1} + b u) \mid x^a y^b \in \mathcal{M} \}$, and to describe dual codes. The framework shows that decreasing norm-trace codes contain one-point AG codes on ${\mathcal{X}}_u$ as a special case and provides self-orthogonality/self-duality criteria. A single-erasure linear repair scheme with bandwidth at most $|{\mathcal{X}}_u| - 1 + (u-1)(r-1)$ is proved, leveraging indicator functions and traces to achieve higher-rate repair than prior schemes for certain higher-rate codes on the extended norm-trace curve.

Abstract

The decreasing norm-trace codes are evaluation codes defined by a set of monomials closed under divisibility and the rational points of the extended norm-trace curve. In particular, the decreasing norm-trace codes contain the one-point algebraic geometry (AG) codes over the extended norm-trace curve. We use Gröbner basis theory and find the indicator functions on the rational points of the curve to determine the basic parameters of the decreasing norm-trace codes: length, dimension, and minimum distance. We also obtain their dual codes. We give conditions for a decreasing norm-trace code to be a self-orthogonal or a self-dual code. We provide a linear exact repair scheme to correct single erasures for decreasing norm-trace codes, which applies to higher rate codes than the scheme developed by Jin, Luo, and Xing (IEEE Transactions on Information Theory {\bf 64} (2), 900-908, 2018) when applied to the one-point AG codes over the extended norm-trace curve.

Figures (4)

• Figure 1: Take $q=3$ and $r=2$. (a) Shows the points of the norm-trace curve $\mathcal{X}: x^4=y^3+y$. Let $\textcolor{red}{\mathcal{M}}$ be the set of monomials whose exponents are the points in (b). The evaluation code ${\rm ev}(\textcolor{red}{\mathcal{M}})$ is an $[ 27, 10, 15 ]$ decreasing norm-trace code over ${\mathbb F}_9$.
• Figure 2: Take $q=2$, $r=4$, and $u=3$. (a) Shows the points of the norm-trace curve $\mathcal{X}_u: x^3=y^8+y^4+y^2+y$. Let $\textcolor{red}{\mathcal{M}}$ be the set of monomials whose exponents are the points in (b). The evaluation code ${\rm ev}(\textcolor{red}{\mathcal{M}})$ is an $[ 32, 12, 12 ]$ decreasing norm-trace code over ${\mathbb F}_{16}$.
• Figure 3: (a) shows the exponents of the set of monomials $\textcolor{red}{\mathcal{M}}$ in $\Delta\left(x^{9}, y^{3}\right)$ of degree at most 4. (b) shows the exponents of $\textcolor{blue}{\mathcal{M}^\complement}$, the complement of $\textcolor{red}{\mathcal{M}}$ on $\mathcal{X}$. By Theorem \ref{['22.03.15']}, the dual code ${\rm ev}\left(\textcolor{red}{\mathcal{M}}\right)^\perp$ is equivalent to the code ${\rm ev}\left(\textcolor{blue}{\mathcal{M}^\complement}\right)$.
• Figure 4: (a) shows the points of the curve $\mathcal{X}: x^u=y^8+y^4+y^2+y$. Let $\textcolor{red}{\mathcal{M}}$ be the set of monomials whose exponents are the points in (b). The evaluation code ${\rm ev}(\textcolor{red}{\mathcal{M}})$ is a self-dual code over ${\mathbb F}_{16}$.

Theorems & Definitions (32)

• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Definition 4.1
• Example 4.2