Source-linked AI summary
Linear Codes over $\mathbb{F}_{q}+u\mathbb{F}_{q}$ associated with Simplicial Complexes, Their Gray Images, and Subfield Codes
Ankit Yadav, Akanksha Tiwari, Ritumoni Sarma
TL;DR
Distance-optimal codes are important for error correction, and simplicial complexes provide a construction approach. This paper builds ring codes and studies their Gray images and subfield codes. It reports distance-optimal families, including a near-Griesmer family, and gives sufficient optimality and minimality conditions.
Problem
Constructing distance-optimal codes is a fundamental coding-theory objective, and the paper addresses constructions from simplicial complexes over Fq + uFq and their subfield codes.
Source: §1 · pp. 1–3Method
The authors construct codes over Fq + uFq from simplicial complexes, determine Lee weight distributions, and study Gray images and corresponding subfield codes.
Source: §1 · pp. 2–3, §6 · p. 26Results
The constructions yield optimal few-weight codes over Fq, including near-Griesmer codes, and sufficient conditions for distance-optimality and minimality of subfield codes.
Source: §4.1 · p. 17, §6 · p. 26Takeaways & Limitations
The paper gives sufficient criteria for distance-optimality and minimality across its Gray-image and subfield-code constructions.
Source: §1 · p. 2, §6 · p. 26
Abstract
from arXiv · showhide
In recent years, simplicial complexes have gained considerable attention as a useful tool for constructing distance-optimal codes over finite fields. In this article, we construct four infinite families of linear codes over the ring $\mathcal{R}=\mathbb{F}_{q}+u\mathbb{F}_{q}$ with $u^2=0$ using simplicial complexes with one or two maximal elements, and completely determine their Lee weight distributions via exponential-sum techniques. By employing a Gray map on $\mathcal{R}$, we obtain infinite families of distance-optimal codes over $\mathbb{F}_{q}$, including a near-Griesmer family, and establish sufficient conditions for their minimality. Furthermore, we investigate the corresponding subfield codes and derive sufficient conditions for their distance-optimality and minimality, yielding infinite families of Griesmer and near-Griesmer codes.
1 Introduction
Distance-optimal codes are a central coding-theory objective, and finite-ring constructions with Gray maps and simplicial complexes offer routes to such codes. This paper constructs codes over Fq + uFq, determines their Lee weight distributions, and studies their Gray images and subfield codes.
1 Introduction
- Distance-optimal codes maximize error-detection and error-correction capabilities for fixed length and dimension, while weight distributions help assess error-correcting capability and error probabilities.Minimal linear codes have applications in secret sharing and secure two-party computation.Source: §1 · pp. 1–2
- Earlier work used simplicial complexes to construct optimal codes over fields and several finite rings, including Fq + uFq with one maximal element.The cited prior Fq + uFq construction determined Lee weight distributions and produced distance-optimal codes over Fq.Source: §1 · p. 2
- The paper constructs four infinite families over Fq + uFq using simplicial complexes with one or two maximal elements, determines their Lee weight distributions, and studies their Gray images and subfield codes.It gives sufficient conditions for distance-optimality and minimality, including for the subfield codes.Source: §1 · pp. 2–3
- The article identifies its study of subfield codes arising from codes over an Fq-algebra for arbitrary q as the first of its kind.The paper is organized around code construction and Lee distributions, Gray images, then subfield codes and their optimality and minimality conditions.Source: §1 · p. 3
2 Preliminaries
The preliminaries establish code and bound terminology, connect Lee weight to Gray images, define the simplicial-complex setting, and present tools used in the constructions and proofs.
2 Preliminaries
- A linear code with parameters [n, k, d] has length n, dimension k, and minimum Hamming distance d; the Griesmer bound defines Griesmer and near-Griesmer codes.A code meeting the Griesmer bound is distance-optimal.Source: §2 · p. 3
2.1 Lee Weight and Gray Map
- The Gray map preserves distance from Lee distance over R to Hamming distance over Fq, so the Lee weight distribution matches that of the Gray image.The map is introduced on ring elements and extended to vectors.Source: §2.1 · p. 4
2.2 Simplicial Complex
- A simplicial complex is described through maximal elements and their supports, with complexes generated by one or two maximal elements denoted by ΔA and ΔA,B.The two-generator complex is the union of the single-generator complexes.Source: §2.2 · p. 4
2.3 Auxiliary Lemmas
- The auxiliary results include sufficient criteria for minimality and distance-optimality of near-Griesmer codes, alongside lemmas used to prove optimality.For a near-Griesmer code with dimension greater than one, the stated criterion requires q to divide d.Source: §2.3 · p. 5
2.4 CD-Construction and Weight Formula
- The CD construction uses a defining set D to form a linear code over R, and its Lee weight is expressed through two Hamming-weight terms.The weight calculation uses orthogonality of nontrivial additive characters.Source: §2.4 · pp. 6–7
3 Lee Weight Distributions
The section constructs four classes of CD-codes from simplicial complexes and determines their Lee weight distributions. The classes have different weight-count bounds and parameter formulas, with subset conditions yielding lower-weight special cases.
3 Lee Weight Distributions
- Across the four classes, defining sets built from simplicial complexes with one or two maximal elements produce CD-codes whose Lee weight distributions are determined.The section introduces these four classes as its focus.Source: §3 · p. 7
- The section also presents a family with at most five weights and length (q^|A| − 1)(q^|B| − 1), and a four-weight special case when A ⊆ B.The corresponding code parameters and Lee weight distributions are stated in the theorem and corollary.Source: §3 · pp. 7, 10
- Theorem 3.3 gives an at-most-three-weight code of length q^2m − q^(|A|+|B|) and cardinality q^2m; when A ⊆ B, Corollary 3.4 gives a two-weight code.The theorem supplies its Lee weight distribution.Source: §3 · pp. 10, 12
- Theorem 3.5 gives an at-most-nine-weight code of length q^|A|(q^m − q^|B| − q^|C| + 2q^|B∩C|) and cardinality q^(m+|A|).Its defining set is formed from ΔA and a complement involving ΔB,C and ΔB∩C; the theorem gives the Lee weight distribution.Source: §3 · p. 12
- Theorem 3.6 gives an at-most-nine-weight code of length q^|A|(q^m − q^|B| − q^|C| + q^|B∩C|) and cardinality q^(m+|A|).Its Lee weight distribution follows from Theorem 3.5.Source: §3 · p. 16
4 Structure of the Gray Image Φ(CD)
Section 4 establishes distance-optimality results for Gray images of the constructed codes and gives sufficient conditions for their minimality.
4.1 Distance-Optimality
- Theorems 4.1, 4.5, and 4.7 give parameters for further Gray-image families and state conditions for distance-optimality.The supplied theorem passages do not provide all of those conditions.Source: §4.1 · pp. 17–18
- Examples report distance-optimal Gray images with parameters, [452, 8, 224], and.The examples associate these parameters with Theorems 4.1, 4.5, and 4.7, respectively.Source: §4.1 · pp. 17–19
- When |A| + |B| > 1, Theorem 4.3 states that the Gray image is both near-Griesmer and distance-optimal.The theorem gives a family of codes over Fq.Source: §4.1 · p. 17
4.2 Minimality
- Theorem 4.9 gives four sufficient conditions for a Gray image to be minimal, including A ⊆ B with |A| > 1 and bounds on set sizes.The other two conditions require q|B| + q|C| < q^m − 1.Source: §4.2 · p. 19
- Examples 4.10–4.12 illustrate minimal Gray images with parameters, [112, 6, 56], and [56, 5, 26].The examples verify the corresponding weight enumerators by Magma.Source: §4.2 · p. 20
5 Subfield Codes
The section defines subfield codes through basis-coordinate and trace constructions, then establishes weight and distance properties for several families. It also gives sufficient conditions for minimality of the resulting subfield codes.
5 Subfield Codes
- A subfield code is formed by replacing generator-matrix entries with their coordinate representations over Fq; Theorem 5.2 gives a trace-based generator construction.For R = Fq + uFq, the paper uses the basis {1, u} and trace a + ub ↦ a + b.Source: §5 · p. 20
- For the family in Theorem 5.4, the subfield code is at most three-weight and near-Griesmer; it is distance-optimal when |A| > 1.The theorem assumes |A| ≤ |B|.Source: §5 · p. 21
- Theorem 5.6 gives a two-weight subfield code that is a Griesmer code and hence distance-optimal.Its parameters are stated in the theorem.Source: §5 · p. 24
- Theorems 5.8 and 5.10 give at-most-five-weight subfield codes, with lengths and dimensions determined by their respective defining sets.The two theorems state different length formulas.Source: §5 · p. 24
5.1 Minimality of Subfield Codes
- Theorem 5.11 gives sufficient minimality conditions for four subfield-code families, with conditions depending on |A|, |B|, |C|, and q^m.The conditions are |A| > 1; |A| + |B| < 2m − 1; or, for either of the last two families, q|B| + q|C| < q^m − 1.Source: §5.1 · p. 25
6 Conclusion
The paper studies four classes of codes over Fq + uFq, determines their Lee weight distributions, and derives optimality and minimality criteria for Gray images and subfield codes. It identifies other defining sets and finite alphabets as directions for future research.
6 Conclusion
- The authors investigate four classes of codes over Fq + uFq built from simplicial complexes with one or two maximal elements and determine their Lee weight distributions.Examples illustrate the theoretical results, and MAGMA is used to verify their parameters.Source: §6 · p. 26
- Gray maps yield families of optimal few-weight codes over Fq, with sufficient criteria for minimality.Source: §6 · p. 26
- The paper derives sufficient conditions for distance-optimality and minimality of the corresponding subfield codes.Source: §6 · p. 26
- Future work may seek other defining sets over Fq + uFq or other finite alphabets that produce infinite families of distance-optimal codes.Source: §6 · p. 26