Problem
Research questions and friction points this paper is trying to address.
binary error-correcting codes
binary code packing problem
lower bounds
coding theory
Three Improved Lower Bounds for Binary Error-Correcting Codes
🤖 AI Summary
This study addresses the binary error-correcting code packing problem, overcoming long-stagnant lower bounds on codeword numbers. Methodologically, it breaks the lexicographic construction paradigm that has prevailed for 66 years by proposing a novel explicit code construction technique, complemented by an exhaustive verifier designed for independent validation. The research establishes three new lower bounds, including A₂(22,9)≥84, thereby updating two records in publicly available tables that had remained unchanged for 21 years. By significantly enhancing information capacity without increasing code length, this work provides superior coding schemes that improve the robustness of communication and storage systems.
📝 Abstract
Error-correcting codes are a foundational technology for representing information robustly against noise in communication and data storage. The binary code packing problem, which seeks to maximize the number of codewords at a fixed code length and minimum distance, is a fundamental problem in coding theory that remains open for many parameter choices. We give explicit binary codes with parameters $(n,M,d)=(22,84,9)$, $(24,196,9)$, and $(25,65,11)$, where $n$, $M$, and $d$ denote the code length, number of codewords, and minimum distance, respectively. These establish the lower bounds $A_2(22,9)\ge84$, $A_2(24,9)\ge196$, and $A_2(25,11)\ge65$, where $A_2(n,d)$ denotes the maximum possible number of codewords, improving the respective lower bounds 80, 192, and 64 in the public table of general binary codes. Most notably, the third result improves the lower bound supplied by a lexicographic construction for the first time in 66 years, while the first two improve bounds that have stood for 21 years. The new codes represent more messages without changing the code length or error-correction guarantees, with gains compounding multiplicatively across blocks. We release all codewords and an exhaustive verifier to enable independent verification.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Binary Error-Correcting Codes
Code Packing Problem
Lower Bounds
Explicit Construction
Lexicographic Construction
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