[Paper Review] Near-Optimal Vector Linear Index Codes For Single Unicast Index Coding Problems with Symmetric Neighboring Interference

Mahesh Babu Vaddi, B. Sundar Rajan|arXiv (Cornell University)|May 28, 2017

Cooperative Communication and Network Coding1 references3 citations

TL;DR

This paper proposes a near-optimal vector linear index code construction for single unicast index coding problems with symmetric neighboring interference (SUICP-SNI), achieving rates within a bounded gap of the theoretical lower bound. It introduces a family of 2-tuples (a,b) such that the rate $ D+1 + \frac{a}{b} $ is achievable over any field, with the gap to the lower bound quantified as $ \frac{K \mod (D+1)}{\lfloor K/(D+1) \rfloor} $, recovering known exact capacity results as special cases.

ABSTRACT

A single unicast index coding problem (SUICP) with symmetric neighboring interference (SNI) has equal number of $K$ messages and $K$ receivers, the $k$th receiver $R_{k}$ wanting the $k$th message $x_{k}$ and having the side-information $\mathcal{K}_{k}=(\mathcal{I}_{k} \cup x_{k})^c,$ where ${I}_k= \{x_{k-U},\dots,x_{k-2},x_{k-1}\}\cup\{x_{k+1}, x_{k+2},\dots,x_{k+D}\}$ is the interference with $D$ messages after and $U$ messages before its desired message. Maleki, Cadambe and Jafar obtained the capacity of this single unicast index coding problem with symmetric neighboring interference (SUICP-SNI) with $K$ tending to infinity and Blasiak, Kleinberg and Lubetzky for the special case of $(D=U=1)$ with $K$ being finite. In our previous work, we proved the capacity of SUICP-SNI for arbitrary $K$ and $D$ with $U= ext{gcd}(K,D+1)-1$. This paper deals with near-optimal linear code construction for SUICP-SNI with arbitrary $K,U$ and $D.$ For SUICP-SNI with arbitrary $K,U$ and $D$, we define a set of $2$-tuples such that for every $(a,b)$ in that set the rate $D+1+\frac{a}{b}$ is achieved by using vector linear index codes over every field. We prove that the set $\mathcal{\mathbf{S}}$ consists of $(a,b)$ such that the rate of constructed vector linear index codes are at most $\frac{K~ ext{mod}~(D+1)}{\left \lfloor \frac{K}{D+1} ight floor}$ away from a known lower bound on broadcast rate of SUICP-SNI. The three known results on the exact capacity of the SUICP-SNI are recovered as special cases of our results. Also, we give a low complexity decoding procedure for the proposed vector linear index codes for the SUICP-SNI.

Motivation & Objective

• To design near-optimal vector linear index codes for single unicast index coding problems with symmetric neighboring interference (SUICP-SNI) for arbitrary K, U, and D.
• To achieve broadcast rates within a quantified gap of the known lower bound on the broadcast rate for SUICP-SNI.
• To generalize previous results on exact capacity for specific U values, such as $ U = \gcd(K, D+1) - 1 $, to a broader class of parameters.
• To provide a low-complexity decoding procedure for the constructed vector linear index codes.

Proposed method

• Define a set $ \mathcal{S} $ of 2-tuples (a,b) such that the rate $ D+1 + \frac{a}{b} $ is achievable via vector linear index codes over any field.
• Construct index codes using AIR (Adjacent Independent Row) matrices to ensure linear independence of adjacent rows over any field.
• Use a recursive structure based on the extended Euclidean algorithm to generate code matrices with controlled row dependencies and interference patterns.
• Design a decoding procedure that leverages side-information and coded symbol combinations to cancel interfering messages, ensuring correct decoding of desired symbols.
• Prove that the constructed codes achieve a rate within $ \frac{K \mod (D+1)}{\lfloor K/(D+1) \rfloor} $ of the known lower bound on the broadcast rate.
• Verify that the construction recovers three known exact capacity results as special cases, including Maleki et al.'s asymptotic capacity and Blasiak et al.'s $ D=U=1 $ result.

Experimental results

Research questions

• RQ1 Can a near-optimal vector linear index code be constructed for SUICP-SNI with arbitrary K, U, and D, achieving a rate close to the theoretical lower bound?
• RQ2 What is the maximum gap between the constructed code rate and the known lower bound on the broadcast rate for SUICP-SNI?
• RQ3 How can the construction be generalized to recover known exact capacity results as special cases?
• RQ4 What is the structure of the code matrices that ensures linear independence of adjacent rows over any field?
• RQ5 Can a low-complexity decoding procedure be designed that enables efficient decoding using only side-information and coded symbols?

Key findings

• The proposed vector linear index codes achieve a rate of $ D+1 + \frac{a}{b} $ for a set of 2-tuples (a,b), which is valid over any finite field.
• The rate of the constructed codes is within $ \frac{K \mod (D+1)}{\lfloor K/(D+1) \rfloor} $ of the known lower bound on the broadcast rate for SUICP-SNI.
• The construction recovers the exact capacity results of Maleki et al. (asymptotic, $ K \to \infty $) and Blasiak et al. ($ D=U=1 $, finite K) as special cases.
• The AIR matrix construction ensures that any $ n $ adjacent rows are linearly independent over any field, enabling robust code design.
• A low-complexity decoding procedure is provided, where each receiver cancels interfering symbols by adding specific coded symbols based on side-information.
• The decoding process is proven to correctly recover the desired message symbol by leveraging the structure of the code and side-information, even under symmetric neighboring interference.