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 | ||||||||

(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 | |||||||||

the k-th code with minimum distance d produced by the lexicographic construction | is given in Figure 2.2 on page | ||||||||||

a| b | the concatenation of a and b | 111 | 010 = 111010 | |||||||||

a^{i} | the concatenation of a with itself i times |
(01)^{3} = 010101 | |||||||||

f_{lexi},
f_{trelli},
f_{state},
f_{decoding} | generating mappings for lexicodes, trellis-oriented codes, state-bounded codes, and decoding-bounded codes respectively | See Section 2.2 | |||||||||

G_{ i}f, 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 | |||||||||

_{d} | the seed code
{0^{d}, 1^{d}} for a particular generalized lexicographic construction |
S_{3} = {000, 111} | |||||||||

R(v),
| the rightmost and leftmost (respectively) significant bit of a bit sequence
v = (v_{1}, v_{2}, v_{3},..., v_{n}) | R(0101) = 4,
| |||||||||

(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 ) | |||||||||

the binary complement of a | = 10101 | ||||||||||

n_{m}, | the length and covering radius (respectively) of k-th code in a G-family | the (7, 3, 4) lexicode has length n_{3}=7 and covering radius = 3 | |||||||||

T(H) | the Tanner graph corresponding to the parity-check matrix H | See Example 3.1 | |||||||||

() | the number of connected components in | For any tree, () = 1 | |||||||||

the Hamming weight (i.e. number of nonzero entries) of a matrix M |
for the rank n identity matrix I_{n} | ||||||||||

_{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 | |||||||||

http://people.bu.edu/trachten