On the Non-Approximate Successive Cancellation Decoding of Binary Polar Codes With Medium Kernels

Polar codes constructed by large kernels can attain better finite length performance than those originating from Arıkan’s
$2\times 2$
kernel. However, the successive cancellation (SC) decoding for these polar codes is impractical even for relatively small kernel size of
$m$
because complexity of the kernel computation grows exponentially with
$m$
. This research shows when
$m>2$
, there exists a large amount of like terms in the kernel computation which yields a ground for facilitating the decoding. By transferring the kernel computation from the probability domain to the likelihood ratio domain (
$l$
-domain), the so- called
$l$
-formula method provides an efficient way to combine the like terms in the kernel computation for kernels up to size 11. However, the
$l$
-formula method becomes intractable for kernel size beyond 11. To further reduce the computational complexity, this paper proposes a
$W$
-formula method which transforms the kernel computation into the probability pair domain (
$W$
-domain). Advanced from the
$l$
-domain, the numerator and denominator of the likelihood ratio are considered separately, which eases the restrictions of combining like terms. The
$W$
-formula method can combine much more like terms resulting in a significant reduction on the number of sub-formulas for medium kernels (
$m\leq 16$
). Furthermore, in the
$W$
-domain, sub-formulas become regular and there exist many common sub-formulas whose computations can be shared. Being able to handle kernels of size up to 16, we show that the
$W$
-formula based SC decoding achieves a significant complexity reduction over the existing non-approximate SC decoding (the
$l$
-formula based SC decoding).