Optimal Scalar Linear Index Codes for One-Sided Neighboring Side-Information Problems

TL;DR

This paper constructs optimal scalar linear index codes for symmetric one-sided neighboring side-information problems, ensuring independence of field size and achieving capacity for any number of messages and antidotes.

Contribution

It introduces matrices with linearly independent adjacent rows to design optimal scalar linear index codes for symmetric one-sided antidote problems.

Findings

Codes are optimal and field-size independent.

Constructed matrices ensure linear independence of adjacent rows.

Applicable for any number of messages and antidotes.

Abstract

The capacity of symmetric instance of the multiple unicast index coding problem with neighboring antidotes (side-information) with number of messages equal to the number of receivers was given by Maleki \textit{et al.} In this paper, we construct matrices of size $ m \times n (m \geq n)$ over $F_q$ such that any $n$ adjacent rows of the matrix are linearly independent. By using such matrices, we give an optimal scalar linear index codes over $F_q$ for the symmetric one-sided antidote problems considered by Maleki \textit{et al.} for any given number of messages and one-sided antidotes. The constructed codes are independent of field size and hence works over every field.