3.7.2 Syndrome Decoding

Suppose that a code vector V = [ v ₀ v ₁ . . . v _{n ?1}] is transmitted. In the presence of noise, the received vector R = [ r ₀ r ₁. . . r _{n ?1}] may not be the same as the transmitted vector. The decoder computes the ( n ? k)-tuple:

(3.30)

where H ^T is the transpose of the parity-check matrix of the ( n, k) linear code and S is the syndrome of R. In the presence of errors:

(3.31)

where R = [ r ₀ r ₁. . . r _{n ?1}] is the received word, V = [ v ₀ v ₁ . . . v _{n ?1}] is the transmitted codeword, and E = [ e ₀ e ₁ . . . e _{n ?1}] is the error pattern.

For ( n, k) systematic linear code with an information vector U = [ u ₀ u ₁ . . . u _{k ?1}], the transmitted codeword V becomes

(3.32)

and equation (3.30) becomes

(3.33)

where

(3.34)

Based on equations (3.14) and (3.34), equation (3.33) can be written as,

(3.35)

It can be seen that the first term corresponds to the received parity-check digit and the remaining terms correspond to the recalculated parity-check...