The Extended Codes of Some Linear Codes
Zhonghua Sun, Cunsheng Ding, Tingfang Chen

TL;DR
This paper introduces nonstandard extensions of linear codes, develops their theory, and applies these methods to various code families, achieving optimal parameters and resolving longstanding open problems.
Contribution
It presents the concept of nonstandard extended codes, develops their theoretical framework, and applies them to multiple code families to find optimal parameters.
Findings
Four families of distance-optimal or dimension-optimal extended codes
Parameters of extended codes for many linear code families are determined
Addresses open problems on nonbinary Hamming code extensions
Abstract
The classical way of extending an linear code is to add an overall parity-check coordinate to each codeword of the linear code . This extended code, denoted by and called the standardly extended code of , is a linear code with parameters , where or . This is one of the two extending techniques for linear codes in the literature. The standardly extended codes of some families of binary linear codes have been studied to some extent. However, not much is known about the standardly extended codes of nonbinary codes. For example, the minimum distances of the standardly extended codes of the nonbinary Hamming codes remain open for over 70 years. The first objective of this paper is to introduce the nonstandardly extended codes of a linear code and develop some general theory for this type of extended…
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Taxonomy
TopicsCoding theory and cryptography · Cancer Mechanisms and Therapy · Multilevel Inverters and Converters
