QC-LDPC Codes From Difference Matrices and Difference Covering Arrays

We give a framework that generalizes LDPC code constructions using transversal designs or related structures such as mutually orthogonal Latin squares. Our constructions offer a broader range of code lengths and codes rates. Similar earlier constructions rely on the existence of finite fields of order a power of a prime, which significantly restricts the functionality of the resulting codes. In contrast, the LDPC codes constructed here are based on difference matrices and difference covering arrays, structures that are available for any order
$a$
, resulting in LDPC codes across a broader class of parameters, notably length
$a(a-1)$
, for all even
$a$
. Such values are not possible with earlier constructions, thus establishing the novelty of these new constructions. Specifically the codes constructed here satisfy the RC constraint and for
$a$
odd, have length
$a^{2}$
and rate
$1-(4a-3)/a^{2}$
, and for
$a$
even, length
$a^{2}-a$
and rate at least
$1-(4a-6)/(a^{2}-a)$
. When 3 does not divide
$a$
, these LDPC codes have stopping distance at least 8. When
$a$
is odd and both 3 and 5 do not divide
$a$
, our construction delivers an infinite family of QC-LDPC codes with minimum distance at least 10. We also determine lower bounds for the stopping distance of the code. Further we include simulation results illustrating the performance of our codes. The BER and FER performance of our codes over AWGN (via simulation) is at least equivalent to codes constructed previously.