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Suslin-Rao Codes : the bridge between pure and applied

Nitin Kenjale, Anuradha S. Garge

arXiv:2610.03746v1cs.ITmath.ITmath.RA

TL;DR

The paper explores how recursive Suslin matrices can be used to construct linear block codes, with explicitly determined parameters. It compares the resulting codes with classical alternatives, including a BCH comparison that illustrates a distance-versus-dimension trade-off.

  • Problem

    The paper explores a less-studied direction: constructing linear block codes from recursive matrix families, particularly Suslin matrices.

    Source: §1 · p. 2

  • Method

    The authors recursively construct generator matrices from Suslin matrices over F2 to define the Suslin-Rao code family and determine its parameters from the recursion index.

    Source: §1 · pp. 2–3

  • Results

    The [32, 16, 6] Suslin-Rao code has slightly lower distance than the [31, 16, 7] BCH code but comparable dimension.

    Source: §5.1 · p. 18

  • Takeaways & Limitations

    The [32, 16, 6] Suslin-Rao code is described as attractive for low-complexity applications despite its slightly lower distance than [31, 16, 7] BCH.

    Source: §5.1 · p. 18

    For r = 5, the cited BCH codes can offer greater distance, while often having lower dimension; the comparison notes this trade-off.

    Source: §5.1 · p. 18

Abstract

from arXiv · show

In the year $1977,$ while developing the theory of modules arising in the context of Serre's problem on projective modules in the paper ``On Stably Free Modules" (see \cite{AS}), A. A. Suslin constructed matrices inductively using unimodular rows. Our aim is to connect these matrices to the current and applicable field of coding theory. To this end, we construct Suslin-Rao codes, by using looking at certain unimodular all one vectors over the finite field $\mathbb{F}_2$. Comparative analysis with standard families such as Hamming, Simplex and BCH codes shows that the Suslin-Rao codes achieve better performance than Hamming and Simplex codes in terms of code rate and error correction capability, though BCH codes may outperform them for large code lengths. Bit error rate (BER) versus signal-to-noise ratio (SNR) simulations further support the practical applicability of these codes in noisy channels. The recursive nature of the construction allows efficient encoder design, making the proposed codes attractive for real-time communication systems and hardware implementation.

1 Introduction

The paper adapts recursive Suslin matrices to construct structured linear block codes, motivated by their algebraic properties. It introduces Suslin-Rao codes, gives parameter and structural analyses, and compares them with selected classical codes.

1 Introduction

  • The construction adapts Hurley and Hurley's codes-from-units method, but the resulting codes are not MDS.The cited method is from “Systems of MDS codes from units and idempotents”.Source: §1 · p. 2
  • Recursive Suslin matrices are presented as an algebraic framework for structured codes with predictable parameters and potentially good error-correcting performance.Source: §1 · p. 2
  • Suslin-Rao codes have length n = 2r, dimension k = 2r−1, and minimum distance d = r or r + 1 according to the parity of positive integer r.The family is defined using Suslin matrices over F2.Source: §1 · p. 2
  • The paper compares the proposed codes with selected classical linear codes using minimum distance and bit-error rate.Source: §1 · p. 3
  • The proposed framework recursively constructs linear-code generator matrices from Suslin matrices.The paper also explicitly determines the resulting codes' parameters from the recursion index.Source: §1 · p. 3
  • The paper examines the codes' algebraic structure, including self-duality conditions and their relation to classical coding-theoretic bounds.Source: §1 · p. 3

2 Preliminaries and Notations

This section establishes binary linear-code notation and recalls generator and parity-check matrices, duality, and bounds. It also connects structured algebraic matrices, including Suslin matrices, to the construction of binary code families.

2 Preliminaries and Notations

  • A binary linear code is a subspace of F2^n; its codewords are vectors, and its minimum distance is the least Hamming distance between distinct codewords.The section also defines Hamming weight, distance, and support.Source: §2 · p. 3
  • A generator matrix has rows forming a basis of the code, while a parity-check matrix characterizes codewords through orthogonality to its rows.The matrices satisfy GHT = 0.Source: §2 · p. 4
  • A code's dual consists of vectors orthogonal to every codeword; a self-dual code equals its dual and therefore has dimension n/2.The section recalls Singleton, Griesmer, and sphere-packing bounds and defines code rate as k/n.Source: §2 · p. 4
  • The paper uses structured algebraic matrices such as Suslin matrices to construct new families of binary linear codes, applying classical algebraic K-theory to coding theory.Source: §2 · p. 5

3 Suslin Matrices

The paper studies recursive Suslin matrices and their specialization over F2 using all-one vectors. It establishes parity-dependent matrix properties and, for odd recursion indices, a relation between block rows.

3 Suslin Matrices

  • The special binary matrices Sr are formed by taking both parameter vectors to be all ones over F2.The paper denotes this specialization by Sr.Source: §3 · p. 6
  • Suslin matrices are recursively constructed, and each row and column of the special Sr has r + 1 ones.The general Sr(v, w) has entries among 0, ±ai, and ±bi, with at most r + 1 nonzero entries per row and column.Source: §3 · p. 6
  • Sr is symmetric for every r; it is nilpotent of index 2 when r is odd and an involution when r is even.It is nonsingular when r is even.Source: §3 · p. 7
  • For even r, distinct rows of Sr are orthogonal over F2.Source: §3 · p. 8
  • For odd r, each row of block B is a linear combination of exactly r rows of block A, and conversely.Source: §3 · p. 8

4 Suslin-Rao Codes

The paper constructs Suslin-Rao linear codes from recursive Suslin matrices over F2 and establishes their parameters and basic properties. It also describes syndrome decoding and the codes’ weight distributions.

4.1 Construction of Suslin-Rao Codes

  • The construction uses a recursive framework for generator matrices, with the special Suslin matrix split into generator and parity-check blocks.The resulting generator matrix A has the form (I2r−1 | Sr−1).Source: §4.1 · p. 10
  • SRCr has length 2r, dimension 2r−1, and minimum distance r+1 for odd r or r for even r.Its code rate is 1/2.Source: §4.1 · pp. 10, 12

4.2 Some Basic Properties of Suslin-Rao Codes

  • For odd r, SRCr is self-dual; the paper also establishes that SRCr satisfies the Singleton, Griesmer, and sphere packing bounds.The self-duality proof shows that the generator and check matrices span the same codeword set.Source: §4.2 · pp. 12–15

4.3 Decoding the Suslin-Rao Code

  • Decoding uses syndrome decoding: the syndrome identifies an error coset, and the decoder selects an error vector before estimating the transmitted codeword.The stated selection rule chooses a coset leader of smallest Hamming weight.Source: §4.3 · p. 15

4.4 Weight Distribution of Suslin-Rao Code

  • For every r, SRCr has A0 = 1 and no nonzero codewords below its minimum weight; its weight distribution is symmetric for odd r but not for even r.The paper attributes the odd-r symmetry to the all-one codeword, which SRCr does not contain when r is even.Source: §4.4 · p. 16

5 Comparative Analysis of Code Performance

The comparisons find that Suslin-Rao codes balance code rate and error-correction distance, while BER simulations show performance comparable to several classical codes at moderate SNR. The results vary by code family and SNR regime, with BCH codes offering a distance advantage in some comparisons.

5.1 Comparison of Parameters

  • At r = 5, Suslin-Rao [32, 16, 6] balances Reed-Muller distance and dimension variants, while BCH codes can offer greater distance, often with reduced dimension.Hadamard codes offer high distance but low dimension; Suslin-Rao is reported to have a higher code rate.Source: Table 2 · p. 17, §5.1 · p. 18
  • For r = 3–7, Suslin-Rao improves minimum distance over similar-length Hamming codes; at r = 3 it matches Simplex distance with more codewords, while at r = 4 Simplex has greater distance but lower dimension.At r = 3, the Suslin-Rao code has 16 codewords versus 8 for Simplex.Source: §5.1 · pp. 17–18

5.2 Rate and Relative Distance Comparison

  • Suslin-Rao offers higher rate than Simplex codes and comparable or better relative distance than Reed-Muller codes of the same order.Source: Table 3 · p. 19, §5.2 · p. 19

5.3 BER Performance Comparison

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