Weight distribution of a class of $p$-ary codes

• URL: http://arxiv.org/abs/2503.19141v1
• Date: Mon, 24 Mar 2025 20:53:04 GMT
• Title: Weight distribution of a class of $p$-ary codes
• Authors: Kaimin Cheng, Du Sheng,
• Abstract summary: We prove all possible weights of codewords in $mathcalC_alpha,beta,beta$, demonstrating that it has at most $p+1$ distinct nonzero weights.<n>We also prove that the dual code $mathcalC_alpha,beta$ is optimal with respect to the sphere packing bound.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Let $p$ be a prime, and let $N$ be a positive integer such that $p$ is a primitive root modulo $N$. Define $q = p^e$, where $e = \phi(N)$, and let $\mathbb{F}_q$ be the finite field of order $q$ with $\mathbb{F}_p$ as its prime subfield. Denote by $\mathrm{Tr}$ the trace function from $\mathbb{F}_q$ to $\mathbb{F}_p$. For $\alpha \in \mathbb{F}_p$ and $\beta \in \mathbb{F}_q$, let $D$ be the set of nonzero solutions in $\mathbb{F}_q$ to the equation $\mathrm{Tr}(x^{\frac{q-1}{N}} + \beta x) = \alpha$. Writing $D = \{d_1, \ldots, d_n\}$, we define the code $\mathcal{C}_{\alpha,\beta} = \{(\mathrm{Tr}(d_1 x), \ldots, \mathrm{Tr}(d_n x)) : x \in \mathbb{F}_q\}$. In this paper, we investigate the weight distribution of $\mathcal{C}_{\alpha,\beta}$ for all $\alpha \in \mathbb{F}_p$ and $\beta \in \mathbb{F}_q$, with a focus on general odd primes $p$. When $\beta = 0$, we establish that $\mathcal{C}_{\alpha,0}$ is a two-weight code for any $\alpha \in \mathbb{F}_p$ and compute its weight distribution. For $\beta \neq 0$, we determine all possible weights of codewords in $\mathcal{C}_{\alpha,\beta}$, demonstrating that it has at most $p+1$ distinct nonzero weights. Additionally, we prove that the dual code $\mathcal{C}_{0,0}^{\perp}$ is optimal with respect to the sphere packing bound. These findings extend prior results to the broader case of any odd prime $p$.

Related papers

• The Communication Complexity of Approximating Matrix Rank [50.6867896228563]
  We show that this problem has randomized communication complexity $Omega(frac1kcdot n2log|mathbbF|)$. As an application, we obtain an $Omega(frac1kcdot n2log|mathbbF|)$ space lower bound for any streaming algorithm with $k$ passes.
  arXiv  Detail & Related papers  (2024-10-26T06:21:42Z)
• A class of ternary codes with few weights [0.0]
  In this paper, we investigate a ternary code $mathcalC$ of length $n$, defined by $mathcalC$ := (textTr) := (textTr(dx), dots, dots, d_n$. Using recent results on explicit evaluations of exponential sums, we determine the Weil bound, and techniques, we show that the dual code of $mathcalC$ is optimal with respect to the Hamming bound.
  arXiv  Detail & Related papers  (2024-10-05T16:15:50Z)
• Efficient Continual Finite-Sum Minimization [52.5238287567572]
  We propose a key twist into the finite-sum minimization, dubbed as continual finite-sum minimization. Our approach significantly improves upon the $mathcalO(n/epsilon)$ FOs that $mathrmStochasticGradientDescent$ requires. We also prove that there is no natural first-order method with $mathcalOleft(n/epsilonalpharight)$ complexity gradient for $alpha 1/4$, establishing that the first-order complexity of our method is nearly tight.
  arXiv  Detail & Related papers  (2024-06-07T08:26:31Z)
• Provably learning a multi-head attention layer [55.2904547651831]
  Multi-head attention layer is one of the key components of the transformer architecture that sets it apart from traditional feed-forward models. In this work, we initiate the study of provably learning a multi-head attention layer from random examples. We prove computational lower bounds showing that in the worst case, exponential dependence on $m$ is unavoidable.
  arXiv  Detail & Related papers  (2024-02-06T15:39:09Z)
• Online Learning of Smooth Functions [0.35534933448684125]
  We study the online learning of real-valued functions where the hidden function is known to have certain smoothness properties. We find new bounds for $textopt_p(mathcal F_q)$ that are sharp up to a constant factor. In the multi-variable setup, we establish inequalities relating $textopt_p(mathcal F_q,d)$ to $textopt_p(mathcal F_q,d)$ and show that $textopt_p(mathcal F
  arXiv  Detail & Related papers  (2023-01-04T04:05:58Z)
• Learning a Single Neuron with Adversarial Label Noise via Gradient Descent [50.659479930171585]
  We study a function of the form $mathbfxmapstosigma(mathbfwcdotmathbfx)$ for monotone activations. The goal of the learner is to output a hypothesis vector $mathbfw$ that $F(mathbbw)=C, epsilon$ with high probability.
  arXiv  Detail & Related papers  (2022-06-17T17:55:43Z)
• On Outer Bi-Lipschitz Extensions of Linear Johnson-Lindenstrauss Embeddings of Low-Dimensional Submanifolds of $\mathbb{R}^N$ [0.24366811507669117]
  Let $mathcalM$ be a compact $d$-dimensional submanifold of $mathbbRN$ with reach $tau$ and volume $V_mathcal M$. We prove that a nonlinear function $f: mathbbRN rightarrow mathbbRmm exists with $m leq C left(d / epsilon2right) log left(fracsqrt[d]V_math
  arXiv  Detail & Related papers  (2022-06-07T15:10:46Z)
• Low-degree learning and the metric entropy of polynomials [44.99833362998488]
  We prove that any (deterministic or randomized) algorithm which learns $mathscrF_nd$ with $L$-accuracy $varepsilon$ requires at least $Omega(sqrtvarepsilon)2dlog n leq log mathsfM(mathscrF_n,d,|cdot|_L,varepsilon) satisfies the two-sided estimate $$c (1-varepsilon)2dlog
  arXiv  Detail & Related papers  (2022-03-17T23:52:08Z)
• Learning low-degree functions from a logarithmic number of random queries [77.34726150561087]
  We prove that for any integer $ninmathbbN$, $din1,ldots,n$ and any $varepsilon,deltain(0,1)$, a bounded function $f:-1,1nto[-1,1]$ of degree at most $d$ can be learned.
  arXiv  Detail & Related papers  (2021-09-21T13:19:04Z)
• Spectral properties of sample covariance matrices arising from random matrices with independent non identically distributed columns [50.053491972003656]
  It was previously shown that the functionals $texttr(AR(z))$, for $R(z) = (frac1nXXT- zI_p)-1$ and $Ain mathcal M_p$ deterministic, have a standard deviation of order $O(|A|_* / sqrt n)$. Here, we show that $|mathbb E[R(z)] - tilde R(z)|_F
  arXiv  Detail & Related papers  (2021-09-06T14:21:43Z)
• Sharper bounds for online learning of smooth functions of a single variable [0.0]
  We show that $opt_1+epsilon(mathcalF_q) = Theta(epsilon-frac12)$, where the constants in the bound do not depend on $q$. We also show that $opt_1+epsilon(mathcalF_q) = Theta(epsilon-frac12)$, where the constants in the bound do not depend on $q$.
  arXiv  Detail & Related papers  (2021-05-30T23:06:21Z)
• A Canonical Transform for Strengthening the Local $L^p$-Type Universal Approximation Property [4.18804572788063]
  $Lp$-type universal approximation theorems guarantee that a given machine learning model class $mathscrFsubseteq C(mathbbRd,mathbbRD)$ is dense in $Lp_mu(mathbbRd,mathbbRD)$. This paper proposes a generic solution to this approximation theoretic problem by introducing a canonical transformation which "upgrades $mathscrF$'s approximation property"
  arXiv  Detail & Related papers  (2020-06-24T17:46:35Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.