Double-Layer Low-Density Parity-Check Codes over Multiple-Input Multiple-Output Channels

Algorithm 1

Modified PEG algorithm.

(1) 𝑞=0.

(2)𝐲=∅

(3) for 𝑙=0 do

(4)  arrange the check node degrees of the

  check node group 𝐠_𝑙 in nondecreasing order

  𝐱_𝑖=[𝑥⁰_𝑖,𝑥¹_𝑖,…,𝑥^𝑟𝑖_𝑖⁻¹]=[𝜌_{⌊𝑛𝑖/𝑟𝑖⌋},𝜌_{⌊𝑛𝑖/𝑟𝑖⌋},…,𝜌_{⌈𝑛𝑖/𝑟𝑖⌉}].

(5)  end for

(6)  for 𝑗=0 to 𝑛−1 do

(7)  if 𝑗≥𝑛−𝑚 then

(8)   𝐬_𝑐={𝑐_{𝑗−(𝑛−𝑚)},…,𝑐_𝑚−1}⧵𝐲.

(9)  else

(10)    𝐬_𝑐={𝑐₀,𝑐₁,…,𝑐_𝑚−1}⧵𝐲.

(11)  end if

(12)  for 𝑘=0 to 𝑀−1 do

(13)   if 𝑘=0 then

(14)    if 𝑗≥𝑛−𝑚 then

      𝐞⁰_𝑣𝑗← edge (𝑐_𝑖,𝑣_𝑗), where 𝐞⁰_𝑣𝑗 is the first edge

   incident to 𝑣_𝑗. This edge corresponds to the “1” in

   the diagonal line of matrix 𝐇^𝑝.

(15)    else

    𝐞⁰_𝑣𝑗← edge (𝑐_𝑖,𝑣_𝑗), where 𝑖=(𝑗+(𝑟₁−𝑠₁))

   mod𝑟₁.

(16)    end if

(17)   else

   expand a subgraph from 𝑣_𝑗 up to depth 𝑙 under

  the current graph setting 𝐧^𝑙_𝑣𝑗 such that 𝐧^𝑙_𝑣𝑗∩𝐬_𝑐≠∅

  but 𝐧_𝑣^𝑙+1_𝑗∩𝐬_𝑐=∅, or the cardinality of 𝐧^𝑙_𝑣𝑗 stops

  increasing, then 𝐞^𝑘_𝑣𝑗← edge(𝑐_𝑖,𝑣_𝑗), where 𝐞^𝑘_𝑣𝑗 is the

  𝑘th edge incident to 𝑣_𝑗 and 𝑐_𝑖 is a check node picked

  from the set 𝐧^𝑙_𝑣𝑗∩𝐬_𝑐 having the lowest check-node

  degree.

(18)   end if

(19)   if the degree of 𝑐_𝑖 equals 𝑥^𝑞_𝑘 then

     Add 𝑐_𝑖 into 𝑌.

      𝐱_𝑖←𝐱_𝑖⧵𝑥^𝑞_𝑘.

      𝑞=𝑞+1.

(20)   end if

     find out which check node group 𝐠_𝑖 includes 𝑐_𝑖.

   𝐬_𝑐←𝐬_𝑐⧵𝐠_𝑖.

(21)   end for

(22)  end for