Concatenated Sum-Rank Codes

TL;DR

This paper introduces the concatenation of a sum-rank code and a Hamming metric code and obtains an asymptotically good sequence of sum-rank codes exceeding the Tsfasman-Vladut-Zink-like bound and the Gilbert-Varshamov-like bound.

Abstract

Sum-rank codes have wide applications in multishot network coding, distributed storage and the construction of space-time codes. Asymptotically good sequences of linearized algebraic geometry sum-rank codes, exceeding the Gilbert-Varshamov-like bound, were constructed in a recent paper published in IEEE Trans. Inf. Theory by E. Berardini and X. Caruso. We call this bound the Tsfasman-Vladut-Zink-like bound. In this paper, we introduce the concatenation of a sum-rank code and a Hamming metric code. Then many sum-rank codes with good parameters, which are better than sum-rank BCH codes, are constructed simply and explicitly. Moreover, we obtain an asymptotically good sequence of sum-rank codes exceeding the Tsfasman-Vladut-Zink-like bound and the Gilbert-Varshamov-like bound.

Figures (8)

• Figure 1: Comparison of Corollary \ref{['cor: asy_bound_d2']} with the GV-like bound for sum-rank codes of matrix size $2 \times 2$ over ${\bf{F}}_{2}$.
• Figure 2: Comparison of Corollary \ref{['cor: asy_bound_d2']} with the TVZ-like bound for sum-rank codes of matrix size $2 \times 2$ over ${\bf{F}}_{9}$.
• Figure 3: Corollary \ref{['cor: asy_bound_d2']} with $p=4$ and $r=4$.
• Figure 4: Comparison of the GV-like bound and Corollary \ref{['cor: from_rs']} for sum-rank codes over ${\bf{F}}_{3}$ of matrix size $2 \times 2$.
• Figure 5: Comparison of the TVZ-like bound and Corollary \ref{['cor: from_rs']} for sum-rank codes over ${\bf{F}}_{4^2}$ of matrix size $4 \times 4$.

Theorems & Definitions (16)

• Theorem 1
• Theorem 2: BGR
• Theorem 3: Asymptotic Gilbert-Varshamov-like bound OPBBerardini
• Theorem 4: TVZ bound on the Hamming metric HP
• Theorem 5: Tsfasman-Vlăduţ-Zink-like bound Berardini
• Theorem 6
• proof
• Theorem 7
• proof
• Theorem 8