7 research outputs found
Polar Codes with Dynamic Frozen Symbols and Their Decoding by Directed Search
A novel construction of polar codes with dynamic frozen symbols is proposed.
The proposed codes are subcodes of extended BCH codes, which ensure
sufficiently high minimum distance. Furthermore, a decoding algorithm is
proposed, which employs estimates of the not-yet-processed bit channel error
probabilities to perform directed search in code tree, reducing thus the total
number of iterations.Comment: Accepted to ITW201
Finite-Length Scaling of Polar Codes
Consider a binary-input memoryless output-symmetric channel . Such a
channel has a capacity, call it , and for any and strictly
positive constant we know that we can construct a coding scheme
that allows transmission at rate with an error probability not exceeding
. Assume now that we let the rate tend to and we ask how
we have to "scale" the blocklength in order to keep the error probability
fixed to . We refer to this as the "finite-length scaling" behavior.
This question was addressed by Strassen as well as Polyanskiy, Poor and Verdu,
and the result is that must grow at least as the square of the reciprocal
of .
Polar codes are optimal in the sense that they achieve capacity. In this
paper, we are asking to what degree they are also optimal in terms of their
finite-length behavior. Our approach is based on analyzing the dynamics of the
un-polarized channels. The main results of this paper can be summarized as
follows. Consider the sum of Bhattacharyya parameters of sub-channels chosen
(by the polar coding scheme) to transmit information. If we require this sum to
be smaller than a given value , then the required block-length
scales in terms of the rate as , where is a positive
constant that depends on and , and .
Also, we show that with the same requirement on the sum of Bhattacharyya
parameters, the block-length scales in terms of the rate like , where is a constant that
depends on and , and .Comment: In IEEE Transactions on Information Theory, 201