CHARACTERISTIC LINEAR SEQUENCES AND THEIR COSET FUNCTIONS
CHARACTERISTIC LINEAR SEQUENCES AND THEIR COSET FUNCTIONS
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DOI：
10.1137/0114079
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发表时间：
1966-01-01
影响因子：
1.9
通讯作者：
GOLD, R
中科院分区：
文献类型：
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作者：
GOLD, R
1. 简介。对应于固定递归关系的二进制序列的向量空间已被广泛研究；最大长度的线性递归序列（m序列）一直是[1]、[2]中专门研究和应用的主题。对应于固定递归关系的所有 m 序列的向量空间由彼此循环相移的序列组成，并且这些相位位置之一具有这样的属性：如果通过每隔一项对序列 h 进行采样，则所得序列再次为 h，即对于所有 i 为 h (2i) h (i)。我们将具有这种不变性的线性递归序列称为特征序列。如果周期P的特征线性序列h的定义域被等价关系划分，则a b 当且仅当存在一个整数k使得a b 。 2k 模 P，则显然 h 在该关系的等价类上是常数。在本文中，我们给出了特征线性序列的建设性确定，并确定了上述等价类上的黄金期特征m序列的值。 2．符号和背景。在本节中，我们回顾并开发了续集中需要的一些材料。设 E 为所有序列的集合，其值在两个元素 GF (2) 的域中；该集合是相对于序列加法和乘法的通常操作的积分域，例如，hi, h E (R) 意味着 (hi 和 (hh)(tc)= 0h (i) h (tc i)。该积分域的单位是那些序列 h E 使得 h (0) 0。我们用 h- 表示单位 h 的逆，并注意 h-1 是使得 hl () 的序列
1. Introduction. Vector spaces of binary sequences corresponding to a fixed recursion relation have been extensively studied; the linear recursive sequences of maximal length (m-sequences) have been the subiect of special study and applicationin [1],[2]. The vector space of all m-sequences cor-responding to a fixed recursion relation consists of sequences which are all cyclic phase shifts of one another and one of these phase positions has the property that if the sequence h is sampled by taking every other term, then the resultant sequence is again h, ie, h (2i) h (i) for all i. We designate the linear recursive sequences having this invariance property as characteristic sequences. If the domain of the characteristic linear sequence h of period P is partitioned by theequivalence relation, a b if and only if there exists an integer k such that a b. 2k modulo P, then it is clear that h is constant on the equivalence classes of this relation. In this note we give a constructive determination of characteristic linear sequences and determine the value of characteristic m-sequences of prime period on the equivalence classes described above.2. Notation and background. In this section we review and develop some material which will be needed in the sequel. Let E be the set of all sequences with values in the field of two elements GF (2); this set is an integral domain with respect to the usual operations of sequence addition and multiplication, eg, hi, h E (R) implies (hi and (hh)(tc)= 0h (i) h (tc i). The units of this integral domain are those sequences h E such that h (0) 0. We denote the inverse of the unit h by h-and note that h-1 is the sequence such that hl ()