Irregular product codes

Abstract

Abstract-We introduce irregular product codes, a class of codes where each codeword is represented by a matrix and the entries in each row (column) of the matrix come from a component row (column) code. As opposed to standard product codes, we do not require that all component row codes nor all component column codes be the same. Relaxing this requirement can provide some additional attractive features such as allowing some regions of the codeword to be more error-resilient, providing a more refined spectrum of rates for finite lengths, and improved performance for some of these rates. We study these codes over erasure channels and prove that for any 0 < < 1, for many rate distributions on component row codes, there is a matching rate distribution on component column codes such that an irregular product code based on MDS codes with those rate distributions on the component codes has asymptotic rate 1 − and can decode on erasure channels having erasure probability < (and having alphabet size equal to the alphabet size of the component MDS codes).

Practical Product Code Construction of Polar Codes

“…Polar codes are inherently recursive in nature, being defined through the n-fold Kronecker product of the polarization kernel T 2 . This structure makes it possible to separate the blocks composing the polar code, describing it as an irregular product code [18], i.e. a product code composed by codes of different rates in the same encoding direction.…”

Symmetric product codes

“…Also, the dimension of the code may depend on more than just the component code dimensions (e.g., it may depend on the fine structure of the codes). In some special cases, however, one can still compute the dimension and define a simple systematic encoding procedure [2].…”

“…Using (2) to rewrite this in terms of |supp(x 1 ) ∪ supp(x 2 )|, we observe that D is upper bounded by…”

“…It is worth noting that the encoding of irregular half-product codes can also be handled more easily than irregular product codes. The difficulty with encoding for irregular product codes is that row and column encoding operations must agree on the bits in the parity-on-parity section of the codeword [2]. But, for half-product codes, this condition is automatically enforced by symmetry.…”

“…while the product code scales like O (n ) 1−(t ) 2 . We note that this calculation neglects the additional detail that n ≈ √ 2n and only considers the stopping sets of minimum size.…”