Some (ink, k) cyclic codes in quasi-cyclic form (Corresp.)

DFT Domain Characterization of Quasi-Cyclic Codes

“…Dimension of an one generator code is given by t i where the summation is over the q-cyclotomic cosets modulo n m where the DFT components of the generator are not all zeros, that is, where the corresponding primary components of the code are nonzero. In [15,25], the dimension of the m-quasi-cyclic code generated by a single generator in blocked polynomial form (g (0) (X), g (1) …”

“…The class of quasicyclic codes contains good codes in the sense of meeting a version of the Gilbert-Varshamov bound [14]. With restrictions on the parameters, quasicyclic codes have been investigated in [1,7,8,9,10,11,19,21,22,24,30]. Quasi-cyclic codes have been studied in terms of circulant matrices in [12] and [13].…”

“…For studying m-quasi-cyclic codes, quite often [1,6,7,8,9,10,11,14,15,20,21,22,23,30] the co-ordinates of a codeword a = (a 0 , a 1 , · · · , a n−1 ) are permuted and blocked as ((a 0 , a m , a 2m , · · · , a ( n m −1)m ), (a 1 , a m+1 , a 2m+1 , · · · , a ( n m −1)m+1 ), · · · , (a m−1 , a 2m−1 , a 3m−1 , · · · , a n−1 )). With this co-ordinate ordering, the generator and parity check matrices (with possibly some redundant rows) can be written as matrices with n m × n m circulant matrices as elements.…”

“…It specializes to cyclic codes with m = 1 resulting in only one block in the codewords and circulant matrices as the generator and parity check matrices. In the recent paper [15], Lally and Fitzpatrick consider codewords in the blocked polynomial form as (a (0) (X), a (1) (X), a (2) . The authors then investigate the structural properties of m-quasi-cyclic codes with the help of Groebner bases of modules over F q [X].…”

“…Since (X n m − 1) ⊆ F q [X] annihilates the code, the code can be seen as an module. Unblocked polynomial form of a codeword can be obtained from the blocked polynomial form of a codeword as a(X) = a (0) (X m ) + Xa (1) (X m ) + X 2 a (2) (X m ) + · · · + X m−1 a (m−1) (X m )). In [23], Tanner gave ways to transform a block circulant binary parity check matrix into a parity check matrix over an extension field by a block wise DFT or linearized polynomial transform.…”