data storage designers

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.

Abstract: Function-correcting codes (FCCs) protect the value of a function $f$ of the message against $t$ errors. In the original formulation of Lenz et al. (2023), the encoding is systematic, and the lower bound of $2t$ on the redundancy relies on this form. Recently, D. Ho (arXiv, 2026) showed, for linear functions and linear encodings, that systematicity can cost redundancy. We begin with the OR function to show that the cost is not confined to the linear setting: a non-systematic encoding attains redundancy $1$ while every systematic encoding needs $2t$. For a linear function $f$ and a fixed linear code $C$, let $d_f$ denote the minimum distance between codewords of messages with different function values. Different generator matrices of $C$ assign different codewords to the messages and can give different values of $d_f$. We study which generator matrix of $C$ gives the largest $d_f$. We give an algorithm that constructs an optimal generator matrix and determines the optimal value as the weight of a codeword in a greedy basis of $C$. We then characterize, in terms of information sets, when a generator matrix in systematic form attains this optimum, and give a necessary condition that is checked on the low-weight codewords of $C$ alone. For two-valued functions, we drop both linearity and systematicity. Using the minimality of initial segments of the simplicial order with respect to Hamming neighbourhoods, we show that a non-systematic $(f,t)$-FCC of length $n$ exists if and only if a condition depending on $f$ only through the size of its smaller preimage holds. For $t=1$, redundancy $1$ is sufficient for every nonconstant two-valued function on $\mathbb{F}_2^k$ with $k\ge 10$. For general $t$, redundancy $1$ suffices for all two-valued functions once $k$ is large enough, and for each $s<2t$ we give an upper bound on the threshold in $k$ beyond which redundancy $s$ suffices.