Linear Programming Bound on Frequency Hopping Sequences

IF 2.9 3区计算机科学 Q3 COMPUTER SCIENCE, INFORMATION SYSTEMS

IEEE Transactions on Information Theory Pub Date : 2025-04-08 DOI:10.1109/TIT.2025.3558949

Xing Liu

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引用次数: 0

Abstract

There are several theoretical bounds on frequency hopping (FH) sequences. Each bound is tight in most cases while not tight in some other cases. Besides, the linear programming bound on FH sequences directly converted from that on error correcting codes can be made to be tighter due to the special structure of FH sequences. In this paper, we first give some properties of FH sequences and derive an inequality relationship between FH sequences and Krawtchouk polynomials. By utilizing those properties of FH sequences and the inequality relationship, we establish a linear programming bound on FH sequences. It is actually a nonlinear programming bound for

$\gcd (H_{m}+1,N)\neq 1$

and

$\sum _{j=0}^{H_{m}}A_{j}\geq q^{H_{m}+1}-q^{\frac {H_{m}+1}{\gcd (H_{m}+1,N)}}-1$

, but not difficult to be solved. It is showed that the linear programming bound is tighter than the Peng-Fan bound (Plotkin bound), the sphere-packing bound, the Singleton bound, and the improved Singleton bound in some cases.

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跳频序列的线性规划界

跳频（FH）序列有几个理论边界。在大多数情况下，每个界都是紧的，而在其他一些情况下则不紧。此外，由于跳频序列的特殊结构，直接由纠错码转换而来的跳频序列的线性规划界可以变得更紧。本文首先给出了FH序列的一些性质，并推导了FH序列与Krawtchouk多项式之间的不等式关系。利用跳频序列的这些性质和不等式关系，建立了跳频序列的线性规划界。它实际上是$\gcd (H_{m}+1,N)\neq 1$和$\sum _{j=0}^{H_{m}}A_{j}\geq q^{H_{m}+1}-q^{\frac {H_{m}+1}{\gcd (H_{m}+1,N)}}-1$的非线性规划界，但求解起来并不困难。证明了线性规划界在某些情况下比Peng-Fan界（Plotkin界）、球填充界、单例界和改进单例界更紧。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

IEEE Transactions on Information Theory 工程技术-工程：电子与电气

CiteScore

5.70

自引率

20.00%

发文量

514

审稿时长

12 months

期刊介绍： The IEEE Transactions on Information Theory is a journal that publishes theoretical and experimental papers concerned with the transmission, processing, and utilization of information. The boundaries of acceptable subject matter are intentionally not sharply delimited. Rather, it is hoped that as the focus of research activity changes, a flexible policy will permit this Transactions to follow suit. Current appropriate topics are best reflected by recent Tables of Contents; they are summarized in the titles of editorial areas that appear on the inside front cover.