Block Tensor Rank of Sum-Rank Metric Codes

2026-06-12

AI Summary

This work investigates the structural complexity of sum-rank metric codes and introduces a novel invariant termed “block tensor rank.” This invariant exhibits inter-block additivity, enabling the decomposition of global complexity into tensor rank computations over individual blocks. Leveraging this framework together with tools from tensor rank theory, linear algebra, and classical coding bounds—specifically the Singleton and Griesmer bounds—the authors derive two new families of lower bounds: the projection bound and the coordinate code bound. These are integrated with established bounds to yield explicit, computable limits. The paper further constructs families of sum-rank codes that attain either the Singleton or Griesmer coordinate code bounds, while demonstrating that certain known code families fall short of the Singleton coordinate code bound, thereby validating the efficacy and novelty of the proposed approach.

Abstract

Sum-rank codes provide a generalized framework for Hamming and rank-metric codes, with codewords represented as tuples of matrices and weight given by the sum of the block ranks. In this paper, we introduce and study a block-tensor-rank invariant for sum-rank metric codes. To each code, we associate its \emph{block tensor rank}: the smallest number of block-simple tensors, namely rank-one matrices supported inside single blocks, whose linear span contains the code. In general, determining the block tensor rank of a sum-rank code is challenging. Our main structural result shows that the block tensor rank decomposes additively across the blocks of the code, thereby reducing its computation to a tensor-rank problem on each block projection. Consequently, we derive two complementary lower bounds on the block tensor rank, referred to as the \emph{projection-wise bound} and the \emph{coordinate-code bound}. Moreover, by combining the coordinate-code bound with the classical Singleton and Griesmer bounds for codes in the Hamming metric, we obtain explicit lower bounds, called the \emph{Singleton coordinate-code bound} and the \emph{Griesmer coordinate-code bound}, respectively. We further construct families of sum-rank codes whose block tensor ranks attain the Singleton or Griesmer coordinate-code bounds. These constructions are based on Hamming-metric codes achieving the corresponding classical bounds. Finally, we show that, in certain cases, the block tensor ranks of two known families of sum-rank codes in the literature do not attain the Singleton coordinate-code bound.

Problem

• sum-rank metric codes
• block tensor rank
• tensor rank
• rank-metric codes
• invariant

Innovation

• block tensor rank
• sum-rank metric codes
• tensor decomposition
• Singleton bound
• Griesmer bound