[Paper Review] Optimal Scalar Linear Codes for Some Classes of The Two-Sender Groupcast Index Coding Problem

A Chinmayananda, B. Sundar Rajan|arXiv (Cornell University)|Apr 11, 2018

Cooperative Communication and Network Coding24 references4 citations

TL;DR

This paper proposes a method to construct optimal scalar linear codes for specific classes of the two-sender groupcast index coding problem (TGICP) by leveraging optimal codes from three disjoint single-sender sub-problems. It introduces the concept of joint extensions of multiple single-sender groupcast index coding problems (SGICPs), identifies a class where optimal codes can be constructed from sub-problem codes, and provides a necessary condition for optimality, achieving the minimum aggregate code length in identified cases.

ABSTRACT

The two-sender groupcast index coding problem (TGICP) consists of a set of receivers, where all the messages demanded by the set of receivers are distributed among the two senders. The senders can possibly have a set of messages in common. Each message can be demanded by more than one receiver. Each receiver has a subset of messages (known as its side information) and demands a message it does not have. The objective is to design scalar linear codes at the senders with the minimum aggregate code length such that all the receivers are able to decode their demands, by leveraging the knowledge of the side information of all the receivers. In this work, optimal scalar linear codes of three sub-problems (considered as single-sender groupcast index coding problems (SGICPs)) of the TGICP are used to construct optimal scalar linear codes for some classes of the TGICP. We introduce the notion of joint extensions of a finite number of SGICPs, which generalizes the notion of extensions of a single SGICP introduced in a prior work. An SGICP $\mathcal{I}_E$ is said to be a joint extension of a finite number of SGICPs if all the SGICPs are disjoint sub-problems of $\mathcal{I}_E$. We identify a class of joint extensions, where optimal scalar linear codes of the joint extensions can be constructed using those of the sub-problems. We then construct scalar linear codes for some classes of the TGICP, when one or more sub-problems of the TGICP belong to the above identified class of joint extensions, and provide some necessary conditions for the optimality of the construction.

Motivation & Objective

• To address the two-sender groupcast index coding problem (TGICP), where messages are distributed across two senders and receivers have arbitrary side information and demands.
• To reduce the aggregate code length by exploiting structural relationships between sub-problems of the TGICP.
• To develop a framework for constructing optimal scalar linear codes using optimal codes from disjoint single-sender sub-problems.
• To introduce and characterize a class of joint extensions of multiple SGICPs that preserve optimality in code construction.
• To establish a necessary condition for the optimality of the constructed codes in specific TGICP classes.

Proposed method

• The TGICP is decomposed into three disjoint single-sender groupcast index coding problems (SGICPs), each with no shared messages between them.
• The notion of joint extension is introduced, where a single problem is formed by combining multiple disjoint SGICPs such that each remains a sub-problem.
• A class of joint extensions is identified where optimal scalar linear codes can be constructed from optimal codes of the constituent SGICPs.
• The construction relies on fitting matrices and rank minimization over finite fields, with the optimal code length determined by the sum of the minimum ranks of the fitting matrices of the sub-problems.
• Matrix completion techniques are used, with specific constructions for the combined fitting matrix of the joint extension, ensuring that the row spaces of the sub-problem completions are preserved in the overall solution.
• A necessary condition for optimality is derived based on the rank relationships between the fitting matrices of the sub-problems and the joint extension.

Experimental results

Research questions

• RQ1 Can optimal scalar linear codes for certain classes of the two-sender groupcast index coding problem be constructed using optimal codes from its single-sender sub-problems?
• RQ2 What structural conditions on the sub-problems allow the optimal code of the joint problem to be derived from the optimal codes of the individual sub-problems?
• RQ3 How can the notion of joint extension of multiple SGICPs be formalized to preserve optimality in code construction?
• RQ4 What is the necessary condition for the constructed code to be optimal in the context of joint extensions?
• RQ5 Under what conditions does the aggregate code length of the joint problem equal the sum of the code lengths of the individual sub-problems?

Key findings

• For a class of TGICP instances where the sub-problems form a joint extension of three disjoint SGICPs, the optimal scalar linear code length is given by the sum of the minimum ranks of the fitting matrices of the sub-problems.
• An optimal code construction is provided for a specific example with $\mathbb{F}_2$, achieving a total code length of 5, which matches the theoretical lower bound.
• The construction is valid when the rank hierarchy $r_1 \geq r_{\{1,2\}} \geq r_2$ holds, and the interaction digraph of the problem is $\mathcal{H}_{62}$.
• The joint extension framework ensures that the optimal completions of the sub-problems can be combined into a globally optimal completion of the joint fitting matrix.
• The necessary condition for optimality is that the row space of the combined fitting matrix must be spanned by the first $r_{\{1,2\}}$ rows of the joint completion, preserving the sub-problem structures.
• The method generalizes prior results on single-sender index coding and extends them to the two-sender groupcast setting with shared message distributions.