Reed-Muller Codes
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Reed-Muller type codes over a combinatorial simplex: an algebraic description
arXiv:2606.02819v1 Announce Type: new Abstract: Given an ordered set $B$ of a finite field, a combinatorial simplex over $B$ is defined as the set of vectors such that the positions of the entries, with respect to $B$, sum up to a fixed integer. CAP codes are Reed-Muller type codes defined over a combinatorial simplex.
Betti Numbers and Higher Weight Spectra of Reed-Muller Codes $RM_q(2,2)$
arXiv:2408.02548v3 Announce Type: replace-cross Abstract: We determine all the Betti numbers of the $q$-ary second order Reed-Muller codes of length $q^2$, and also of the elongations of matroids associated to these codes. We then use it to determine the higher weight spectra of these codes. As a special case, we recover some results of Kaplan and Matei about counting certain curves over finite fields with prescribed rational intersection points.
Reed-Muller Codes for Joint Random and Stuck-At Error Correction
arXiv:2605.21727v2 Announce Type: replace Abstract: Block codes are considered for improving the reliability of messages stored in a computer memory with both stuck-at defects and random errors. It is assumed that the side information about the state of the defects is available to the encoder, but not to the decoder. A novel recursive construction of a set of masks is developed such that it can satisfy any $s$ stuck-at errors in a $2^m$ binary sequence, when $s \leq m$. We prove that the...
Implementation and Optimization of HQC Decoding on NPU-Integrated Devices
arXiv:2606.01968v1 Announce Type: new Abstract: Hamming Quasi-Cyclic (HQC) has been selected by NIST for standardization as an additional code-based key-encapsulation mechanism, providing algorithmic diversity alongside lattice-based post-quantum cryptography. Efficient deployment of HQC on mobile and embedded platforms, however, requires careful optimization of its decoding procedure, whose Reed-Muller and Reed-Solomon components dominate the computational cost.
A New Class of Linear Codes
arXiv:2401.07986v3 Announce Type: replace Abstract: Let $n$ be a prime power, $r$ be a prime with $r\mid n-1$, and $\varepsilon\in (0,1/2)$. Using the theory of multiplicative character sums and superelliptic curves, we construct new codes over $\mathbb F_r$ having length $n$, relative distance $(r-1)/r+O(n^{-\varepsilon})$ and rate $n^{-1/2-\varepsilon}$. When $r=2$, our binary codes have exponential size when compared to all previously known families of linear and non-linear codes with...
On the Weight Distribution of Concatenated Code Ensemble Based on the Plotkin Construction
Computer Science > Information Theory [Submitted on 29 Aug 2025 (v1), last revised 30 May 2026 (this version, v2)] Title:On the Weight Distribution of Concatenated Code Ensemble Based on the Plotkin Construction View PDF HTML (experimental)Abstract:In this note, we reveal a relation between the weight distribution of a concatenated code ensemble based on the Plotkin construction and those of its component codes. The relation may find applications in the calculation of the ensemble weight...