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In coding theory, the weight enumerator polynomial of a binary linear code specifies the number of words of each possible Hamming weight. Let C ⊂ F 2 n {\displaystyle C\subset \mathbb {F} _{2}^{n}} be a binary linear code of length n {\displaystyle n} .
A special case of constant weight codes are the one-of-N codes, that encode bits in a code-word of bits. The one-of-two code uses the code words 01 and 10 to encode the bits '0' and '1'. A one-of-four code can use the words 0001, 0010, 0100, 1000 in order to encode two bits 00, 01, 10, and 11.
LDPC codes have no limitations of minimum distance, [35] that indirectly means that LDPC codes may be more efficient on relatively large code rates (e.g. 3/4, 5/6, 7/8) than turbo codes. However, LDPC codes are not the complete replacement: turbo codes are the best solution at the lower code rates (e.g. 1/6, 1/3, 1/2). [36] [37]
The definition of an A p weight and the reverse Hölder inequality indicate that such a weight cannot degenerate or grow too quickly. This property can be phrased equivalently in terms of how much the logarithm of the weight oscillates: (a) If w ∈ A p, (p ≥ 1), then log(w) ∈ BMO (i.e. log(w) has bounded mean oscillation).
The RM(0, m) code is the repetition code of length N =2 m and weight N = 2 m−0 = 2 m−r. By 1 (,) = and has weight 1 = 2 0 = 2 m−r. The article bar product (coding theory) gives a proof that the weight of the bar product of two codes C 1, C 2 is given by
Formally, a parity check matrix H of a linear code C is a generator matrix of the dual code, C ⊥. This means that a codeword c is in C if and only if the matrix-vector product Hc ⊤ = 0 (some authors [1] would write this in an equivalent form, cH ⊤ = 0.) The rows of a parity check matrix are the coefficients of the parity check equations. [2]
A tolerance interval (TI) is a statistical interval within which, with some confidence level, a specified sampled proportion of a population falls. "More specifically, a 100×p%/100×(1−α) tolerance interval provides limits within which at least a certain proportion (p) of the population falls with a given level of confidence (1−α)."
The Power of 10 Rules were created in 2006 by Gerard J. Holzmann of the NASA/JPL Laboratory for Reliable Software. [1] The rules are intended to eliminate certain C coding practices that make code difficult to review or statically analyze.