[Paper Review] On The Number of Optimal Linear Index Codes For Unicast Index Coding Problems

R. Kavitha, Niranjana Ambadi+1 more|arXiv (Cornell University)|Oct 10, 2015

Cooperative Communication and Network Coding13 references4 citations

TL;DR

This paper proposes a novel matrix-based transfer function approach to characterize optimal linear index codes for unicast index coding problems, enabling efficient computation of the minimum number of optimal codes and identifying those with minimum-maximum error probability. By reducing the problem to equivalent network coding with smaller matrices, it establishes a tight lower bound on the number of optimal codes and provides a criterion to select codes that minimize the worst-case decoding error rate across receivers.

ABSTRACT

An index coding problem arises when there is a single source with a number of messages and multiple receivers each wanting a subset of messages and knowing a different set of messages a priori. The noiseless Index Coding Problem is to identify the minimum number of transmissions (optimal length) to be made by the source through noiseless channels so that all receivers can decode their wanted messages using the transmitted symbols and their respective prior information. Recently, it is shown that different optimal length codes perform differently in a noisy channel. Towards identifying the best optimal length index code one needs to know the number of optimal length index codes. In this paper we present results on the number of optimal length index codes making use of the representation of an index coding problem by an equivalent network code. Our formulation results in matrices of smaller sizes compared to the approach of Kotter and Medard. Our formulation leads to a lower bound on the minimum number of optimal length codes possible for all unicast index coding problems which is met with equality for several special cases of the unicast index coding problem. A method to identify the optimal length codes which lead to minimum-maximum probability of error is also presented.

Motivation & Objective

• To determine the number of optimal linear index codes for unicast index coding problems, which is essential for selecting the best-performing code in noisy environments.
• To develop a more efficient matrix formulation than Kotter and Medard's approach, reducing matrix sizes and improving computational feasibility.
• To identify optimal linear index codes that minimize the maximum number of transmissions used by any receiver, thereby minimizing the worst-case probability of error.
• To extend existing results on optimal index coding length to include error performance analysis and code selection criteria.

Proposed method

• Representing the unicast index coding problem as an equivalent network coding problem using input-mixing, transfer, and output-mixing matrices.
• Partitioning the transfer matrix into submatrices corresponding to side information and coded transmissions to analyze decoding complexity.
• Using a transfer matrix approach with smaller component matrices compared to Kotter and Medard's formulation to simplify analysis.
• Defining a set $ S(c) $ of transmission matrices $ T $ for optimal length $ c $, where each $ T $ corresponds to a valid linear index code.
• Deriving a criterion to select $ T \in S(c) $ that minimizes the maximum number of non-zero entries in any row of the $ B_{BC} $ matrix, which corresponds to minimizing the maximum number of transmissions used by any receiver.
• Forming the index code by extracting every $ n $-th row of the $ F_{BC} $ matrix, which represents the broadcasted coded symbols.

Experimental results

Research questions

• RQ1 What is the minimum number of optimal linear index codes possible for any single unicast index coding problem?
• RQ2 How can we identify the optimal linear index code that minimizes the maximum number of transmissions used by any receiver, thereby reducing the worst-case error probability?
• RQ3 Can we derive a lower bound on the number of optimal codes that is tight for specific classes of unicast index coding problems?
• RQ4 How does the structure of the transfer matrix and its submatrices relate to the number and performance of optimal index codes?

Key findings

• A lower bound on the number of optimal linear index codes is derived, which is tight for single uniprior single unicast, single uniprior unicast, and single unicast uniprior problems.
• The number of optimal linear index codes is determined by the size of the set $ S(c) $, which corresponds to valid transmission matrices $ T $ of optimal length $ c $.
• For the single unicast problem with $ m = n = 3 $, three optimal codes were found, each using at most 2 transmissions per receiver, and all exhibited identical worst-case bit error probability (BEP) performance.
• In the $ m = n = 4 $ case, a code with $ t_{\text{max}}(T) = 2 $ achieved better worst-case BER performance than a code with $ t_{\text{max}}(T) = 3 $, confirming the effectiveness of the proposed selection criterion.
• The proposed method reduces matrix complexity compared to Kotter and Medard’s approach, enabling more efficient analysis of optimal index codes.
• The matrix formed by taking every $ n $-th row of $ F_{BC} $ yields the corresponding index code, and the number of non-zero entries in each row of $ B_{BC} $ determines the number of transmissions used by each receiver.