Manan Bhandari received his MS-Dual Degree in Electronics and Communication Engineering. His research work was supervised by Dr. Lalitha Vadlamani. Here’s a summary of Manan’s M.S thesis, On polar codes with product kernels and polarisation of RM codes as explained by him:
Today communication technology relies on state of the art networks, which is led by 5G. Polar codes are one such class of codes known to achieve very high speed in 5G. Any message that needs to be transferred from a noisy channel requires encoding and decoding to keep the message intact and not let the noise corrupt it. Using polar codes, transmission can be done at a rate arbitrarily close to capacity and decoded at the receiver with very small probability of error.
These codes, introduced by Arikan, achieve the capacity of arbitrary binary-input discrete memoryless channel W under successive cancellation decoding. Based on the idea of channel polarization, these codes have simple structure, good performance and are easy to implement with simple encoding and decoding algorithms.
For any such channel having capacity I(W) and for any coding scheme allowing transmission at rate R , the scaling exponent is a parameter that characterizes how fast the gap to capacity decreases as a function of code length N for a fixed probability of error. Scaling exponent for small-sized kernels up to L=8 has been exhaustively found, but as kernel size increases, the complexity to find this exponent increases.
In my work, we find a simple solution to reduce complexity of finding the scaling exponent for a specific type of kernel. We consider product kernels TL obtained by taking Kronecker product of component kernels and derive the properties of polarizing product kernels relating to the number of product kernels, self-duality and partial distances in terms of the respective properties of the smaller component kernels.
Subsequently, the polarization behavior of component kernel Tl is used to calculate the scaling exponent of TL which is double the size of Tl and has lower complexity as compared to the basic method. Using this method, we show that the scaling exponent of T10 is 3.942. Further, we employ a heuristic approach to construct a good kernel of L=14 from the kernel having size l=8 having the best scaling exponent and find T14 is 3.485. Plotting all these values, we make some interesting observations about the nature of this parameter as kernel size increases.
Further, the work of my thesis which focuses on finding the scaling exponent for a specific type of kernel and can be extended to any general kernel. This will result in calculating scaling exponent much efficiently. Another interesting problem to solve is to find if the scaling exponent for modified polar codes like polar subcodes or RM codes exists and, if so, finding those values.
RM codes are known to polarize as well, and an algorithm to find the basis matrix of any RM(m,r) codes are presented in my work, which leads to the generator matrix of these codes needed for encoding. Another category of codes that are very similar to RM codes are twin codes, and it is conjectured that under certain conditions, twin codes are equal to RM codes for binary symmetric channels (BSC). We give a detailed proof of them being the same till m=4 and conclude the proof for a particular case of any general m.