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

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$\ensuremath{\mathcal{L}_{k}^{d}}$ the k-th code with minimum distance d produced by the lexicographic construction $\ensuremath{\mathcal{L}_{4}}^3$ is given in Figure 2.2 on page

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$\langle$ a| b $\rangle$ the concatenation of a and b $\langle$ 111 | 010 $\rangle$ = 111010

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aⁱ the concatenation of a with itself i times (01)³ = 010101

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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

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G_i $\left(\vphantom{ f,\mathbb{C} }\right.$ f, $\mathbb {C}$ $\left.\vphantom{ f,\mathbb{C} }\right)$ 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 $\mathbb {C}$ using the generating mapping f (^.) ${\mathfrak{G}}_{\,3} \left( \ensuremath{f_{\mbox{\small lexi}}},\mathbb{S}_3 \right)$ is the linear code in Table 2.2 on page

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$\mathbb {S}$ _d the seed code {0^d, 1^d} for a particular generalized lexicographic construction S₃ = {000, 111}

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R(v), $\ensuremath{\operatorname{L}}(v)$ the rightmost and leftmost (respectively) significant bit of a bit sequence v = (v₁, v₂, v₃,..., v_n) R(0101) = 4, $\ensuremath{\operatorname{L}}(0101)=2$

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$\kappa_{v}^{}$ (l ) or $\kappa$ (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 $\kappa_{{0110}}^{}$ (0101) = 0011 for code $\mathbb {C}$ in Figure 1.1b (page )

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$\bar{{a}}$ the binary complement of a $\overline{{01010}}$ = 10101

Pages ,
n_m, $\rho_{m}^{}$ 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 $\rho_{3}^{}$ = 3

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

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$\omega$ ( $\cal {G}$ ) the number of connected components in $\cal {G}$ For any tree, $\omega$ ( $\cal {G}$ ) = 1

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$\ensuremath{\operatorname{wt}}(M)$ the Hamming weight (i.e. number of nonzero entries) of a matrix M $\ensuremath{\operatorname{wt}}(I_n)=n$ for the rank n identity matrix I_n

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$\cal {E}$ _n the even-weight (n, n - 1, 2) code, whose parity-check matrixs consists of a single all-1 vector $\cal {E}$ ₂ has vectors 00 and 11

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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	$\ensuremath{\mathcal{L}_{k}^{d}}$	the k-th code with minimum distance d produced by the lexicographic construction	$\ensuremath{\mathcal{L}_{4}}^3$ is given in Figure 2.2 on page
Page	$\langle$ a\| b $\rangle$	the concatenation of a and b	$\langle$ 111 \| 010 $\rangle$ = 111010
Page	aⁱ	the concatenation of a with itself i times	(01)³ = 010101
Page	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
Page	G_i $\left(\vphantom{ f,\mathbb{C} }\right.$ f, $\mathbb {C}$ $\left.\vphantom{ f,\mathbb{C} }\right)$ 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 $\mathbb {C}$ using the generating mapping f (^.)	${\mathfrak{G}}_{\,3} \left( \ensuremath{f_{\mbox{\small lexi}}},\mathbb{S}_3 \right)$ is the linear code in Table 2.2 on page
Page	$\mathbb {S}$ _d	the seed code {0^d, 1^d} for a particular generalized lexicographic construction	S₃ = {000, 111}
Page	R(v), $\ensuremath{\operatorname{L}}(v)$	the rightmost and leftmost (respectively) significant bit of a bit sequence v = (v₁, v₂, v₃,..., v_n)	R(0101) = 4, $\ensuremath{\operatorname{L}}(0101)=2$
Page	$\kappa_{v}^{}$ (l ) or $\kappa$ (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	$\kappa_{{0110}}^{}$ (0101) = 0011 for code $\mathbb {C}$ in Figure 1.1b (page )
Page	$\bar{{a}}$	the binary complement of a	$\overline{{01010}}$ = 10101
Pages ,	n_m, $\rho_{m}^{}$	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 $\rho_{3}^{}$ = 3
Page	T(H)	the Tanner graph corresponding to the parity-check matrix H	See Example 3.1
Page	$\omega$ ( $\cal {G}$ )	the number of connected components in $\cal {G}$	For any tree, $\omega$ ( $\cal {G}$ ) = 1
Page	$\ensuremath{\operatorname{wt}}(M)$	the Hamming weight (i.e. number of nonzero entries) of a matrix M	$\ensuremath{\operatorname{wt}}(I_n)=n$ for the rank n identity matrix I_n
Page	$\cal {E}$ _n	the even-weight (n, n - 1, 2) code, whose parity-check matrixs consists of a single all-1 vector	$\cal {E}$ ₂ has vectors 00 and 11