ECC dictionary M

MacWilliams' identity:

If C is a linear code over GF(q) then the MacWilliams identity states that

A_C^⊥(x,y) = |C|^-1 A_C(x+y,(q-1)x-y),

where C^⊥ denotes the dual code of C.

minimum distance:

A code C has minimum distance

d = min_{c,c' in C, c ≠ c'} wt(c-c'),

where wt is the Hamming weight of a codeword. If C is linear then

d = min_{c in C, c ≠ 0} wt(c).

maximum distance separable:

Lemma: Any linear code is permutation equivalent to a code which is in standard form.

A code C whose parameters satisfy k+d=n+1 is called maximum distance separable or MDS. Such codes, when they exist, are in some sense best possible.