[Paper Review] Sixteen New Linear Codes With Plotkin Sum

Fernando Hernando, Diego Ruano|ArXiv.org|Apr 22, 2008

Coding theory and cryptography3 references3 citations

TL;DR

This paper presents 16 new linear codes over finite fields F₃ and F₄ using the Plotkin sum construction (u|u+v), which combines two known codes to generate longer codes with improved or explicitly constructed minimum distances. Three codes over F₄ exceed previously known lower bounds on minimum distance, while thirteen others provide the first explicit constructions for codes that were previously only known to exist up to bound.

ABSTRACT

Sixteen new linear codes are presented: three of them improve the lower bounds on the minimum distance for a linear code and the rest are an explicit construction of unknown codes attaining the lower bounds on the minimum distance. They are constructed using the Plotkin sum of two linear codes, also called $(u|u+v)$ construction. The computations have been achieved using an exhaustiv search.

Motivation & Objective

• To construct new linear codes with improved or explicitly realized minimum distance bounds using the Plotkin sum method.
• To demonstrate that the Plotkin sum construction remains a viable tool for improving bounds on minimum distance in linear codes.
• To provide explicit constructions for previously unknown codes that meet the theoretical lower bounds on minimum distance.
• To show that a significant portion of codes listed in the database [3] can be generated via Plotkin sum, often more simply than original constructions.
• To encourage future code construction by suggesting the use of Plotkin sum to extend known codes to longer lengths with better distance properties.

Proposed method

• Utilized the Plotkin sum construction: C = {(u, u+v) | u ∈ C₁, v ∈ C₂}, where C₁ and C₂ are linear codes of length n over F_q.
• Applied the parameter formula: the resulting code has parameters [2n, k₁+k₂, min{2d₁, d₂}].
• Performed an exhaustive search over known codes from the database [3] for lengths n ≤ 121 (for q=4), focusing on even-length codes.
• Compared the minimum distance of the Plotkin sum with the known bounds for length 2n in [3] to identify improvements or explicit constructions.
• Used the computer algebra system Magma to implement and verify the construction of specific codes, including shortening operations.
• Evaluated the effectiveness of the Plotkin sum by comparing how many codes in [3] could be reconstructed via this method.

Experimental results

Research questions

• RQ1 Can the Plotkin sum construction still yield new linear codes that improve known lower bounds on minimum distance?
• RQ2 Can the Plotkin sum be used to provide explicit constructions for codes whose existence was previously only known via bounds?
• RQ3 How many codes listed in the database [3] can be generated using the Plotkin sum, and is this method simpler than original constructions?
• RQ4 Does the Plotkin sum construction offer a systematic way to generate longer codes with enhanced minimum distance from shorter known codes?
• RQ5 What is the practical impact of using Plotkin sum in the context of modern code construction and bound improvement?

Key findings

• Three new [126,95,12] and [128,97,12] codes over F₄ improve the lower bound on minimum distance from 11 to 12.
• A [127,96,≥12] code is obtained via shortening of the [128,97,12] code, improving the lower bound from 11 to 12.
• Three new [124,78,16] and [126,79,16] codes over F₃ provide the first explicit constructions for codes meeting the known lower bound on minimum distance.
• Ten new [104,63,16], [106,65,16], [108,67,16], [122,91,12], [124,93,12], and [125,94,≥12] codes over F₄ offer explicit constructions for codes previously only known to exist up to bound.
• The Plotkin sum construction recovers 2,676 of 16,512 (16.2%) even-length codes over F₂, 1,681 of 14,762 (11.38%) over F₃, and significant portions over F₄, F₅, F₇, F₈, F₉, indicating broad applicability.
• For several codes, the Plotkin sum provides a simpler construction path than the original methods involving multiple shortenings, puncturings, or parity checks.