The Gram matrix is a classical object formed from the pairwise inner products of a collection of vectors, with fundamental roles in functional analysis, statistics, combinatorics, and coding theory. In the realm of sequence design, maximum-length sequences (m-sequences) are among the most fundamental classes of sequences, traditionally characterized by their span, decimation, shift-and-add, balance, run, and ideal autocorrelation properties. In this paper, we bridge the two foundational concepts by uncovering novel structural features of m-sequences through the lens of a family of Gram matrices. Specifically, for each $1 \le t \le 2^n - 1$, we extract $n$ consecutive subsequences of length $t$ from an m-sequence of period $2^n - 1$, construct their corresponding $n \times n$ Gram matrix, and investigate its rank, denoted by $r_n(t)$. Utilizing semilinear representation of Galois groups and Bézoutian of polynomials, we derive an explicit formula for $r_n(t)$ for all $t$, thereby establishing the complete rank distribution of these Gram matrices. Notably, we prove that full rank is attained for approximately half of the admissible values of $t$. We further uncover the intricate dynamics of $r_n(t)$: rank-deficient states are strictly unstable (i.e., $r_n(t) < n$ implies $r_n(t+1) \ne r_n(t)$), whereas the full-rank state exhibits strong persistence, remaining at $n$ over a nontrivial interval of consecutive values of $t$. Altogether, our results fully characterize both the global rank distribution and the local dynamics of rank function, as invariant of m-sequences. As an application, our findings completely determine the hull distribution of the family of punctured cyclic simplex codes.

翻译：格拉姆矩阵是由向量集合的成对内积构成的经典对象，在泛函分析、统计学、组合学和编码理论中具有基础性作用。在序列设计领域，最大长序列（m序列）是最基本的序列类别之一，传统上通过其跨度、抽取、移位相加、平衡性、游程和理想自相关性质来刻画。本文通过揭示一类格拉姆矩阵的新结构特征，将这两个基础概念联系起来。具体地，对于每个$1 \le t \le 2^n - 1$，我们从周期为$2^n - 1$的m序列中提取$n$个长度为$t$的连续子序列，构造对应的$n \times n$格拉姆矩阵，并研究其秩$r_n(t)$。利用伽罗瓦群的半线性表示和多项式的贝佐特行列式，我们推导出所有$t$下$r_n(t)$的显式公式，从而建立了这些格拉姆矩阵的完整秩分布。值得注意的是，我们证明约一半的可取$t$值能达到满秩。进一步地，我们揭示了$r_n(t)$的复杂动态性质：缺秩状态严格不稳定（即$r_n(t) < n$蕴含$r_n(t+1) \ne r_n(t)$），而满秩状态则表现出强持续性，在多个连续$t$值的非平凡区间内保持为$n$。总体而言，我们的结果完整刻画了秩函数的全局分布与局部动态，作为m序列的不变量。作为应用，这些发现完全确定了穿刺循环单纯码族的壳分布。