Encoder for quasi-cyclic low-density parity-check codes over subfields using fourier transform

A data storage device comprising:

• a non-volatile memory;
• and
• a quasi-cyclic low-density parity-check (QC-LDPC) encoder configured to encode data for storage into the non-volatile memory, the QC-LDPC encoder including:
• a Fourier transform circuit configured to receive an input message and to generate a transformed message based on the input message, wherein the transformed message includes leading symbols with indices corresponding to leading elements of cyclotomic cosets of a finite field with respect to a subfield;
• and
• a matrix multiplier circuit configured to multiply the leading symbols of the transformed message by leading symbols of a transformed LDPC generator matrix to generate leading symbols of transformed parity symbols associated with an LDPC codeword,
• wherein the QC-LDPC encoder is configured to provide the leading symbols of the transformed parity symbols to an inverse Fourier transform circuit to generate parity information of the LDPC codeword.