# Notation

 Introduced Expression Denotes Example Page [n, M, d] code a code with length n containing M vectors of minimum Hamming distance d from each other See Figure 1.1a on page Page (n, k, d ) code a linear code of length n representing a k dimensional subspace of vectors with minimum Hamming distance d from each other See Figure 1.1b on page Page the k-th code with minimum distance d produced by the lexicographic construction is given in Figure 2.2 on page Page a| b the concatenation of a and b 111  |  010 = 111010 Page ai the concatenation of a with itself i times (01)3 = 010101 Page flexi, ftrelli, fstate, fdecoding generating mappings for lexicodes, trellis-oriented codes, state-bounded codes, and decoding-bounded codes respectively See Section 2.2 Page G if, or G i or G the Generalized Lexicographic Construction; if arguments are supplied, they refer to the i-th iteration of the construction seeded by code using the generating mapping f ( . ) is the linear code in Table 2.2 on page Page d the seed code {0d, 1d} for a particular generalized lexicographic construction S3 = {000, 111} Page R(v), the rightmost and leftmost (respectively) significant bit of a bit sequence v = (v1, v2, v3,..., vn) R(0101) = 4, Page (l ) or (l ) the companion of coset leader l under v; i.e. the coset leader of the coset containing l + v; v is omitted in context (0101) = 0011 for code in Figure 1.1b (page ) Page the binary complement of a = 10101 Pages , nm, the length and covering radius (respectively) of k-th code in a G-family the (7, 3, 4) lexicode has length n3=7 and covering radius = 3 Page T(H) the Tanner graph corresponding to the parity-check matrix H See Example 3.1 Page () the number of connected components in For any tree, () = 1 Page the Hamming weight (i.e. number of nonzero entries) of a matrix M for the rank n identity matrix In Page n the even-weight (n, n - 1, 2) code, whose parity-check matrixs consists of a single all-1 vector 2 has vectors 00 and 11

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