Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Polar codes form a very porful family of codes with a low complexity decoding algorithm that attains many information theoretic limits in error correction and source coding. These codes are closely related to Reed-Muller codes because both can be described with the same algebraic formalism, namely they are generated by evaluations of monomials. Hover, finding the right set of generating monomials for a polar code which optimises the decoding performances is a nontrivial task and is channel dependent. The purpose of this paper is to reveal some universal properties of these monomials.  will namely prove that there is a way to define a nontrivial (partial) order on monomials so that the monomials generating a polar code devised for a binary-input symmetric channel always form a decreasing set.  call such codes decreasing monomial codes. The fact that polar codes are decreasing monomial codes turns out to have rather deep consequences on their structure. Indeed,  show that decreasing monomial codes have a very large permutation group by proving that it contains a group called lor triangular affine group. Furthermore, the codewords of minimum ight correspond exactly to the orbits of the minimum ight codewords that are obtained from evaluations of monomials of the generating set. In particular, it gives an efficient way of counting the number of minimum ight codewords of a decreasing monomial code and henceforth of a polar code.
['Magali Bardet', 'Vlad Dragoi', 'Ayoub Otmani', 'Jean-Pierre Tillich']
I am writing a research paper on a idea Algebraic properties of polar codes from a new polynomial formalism
[{"from": "human", "value": "I am writing a research paper on a idea Algebraic properties of polar codes from a new polynomial formalism"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Polar codes form a very porful family of codes with a low complexity decoding algorithm that attains many information theoretic limits in error correction and source coding. These codes are closely related to Reed-Muller codes because both can be described with the same algebraic formalism, namely they are generated by evaluations of monomials. Hover, finding the right set of generating monomials for a polar code which optimises the decoding performances is a nontrivial task and is channel dependent. The purpose of this paper is to reveal some universal properties of these monomials.  will namely prove that there is a way to define a nontrivial (partial) order on monomials so that the monomials generating a polar code devised for a binary-input symmetric channel always form a decreasing set.  call such codes decreasing monomial codes. The fact that polar codes are decreasing monomial codes turns out to have rather deep consequences on their structure. Indeed,  show that decreasing monomial codes have a very large permutation group by proving that it contains a group called lor triangular affine group. Furthermore, the codewords of minimum ight correspond exactly to the orbits of the minimum ight codewords that are obtained from evaluations of monomials of the generating set. In particular, it gives an efficient way of counting the number of minimum ight codewords of a decreasing monomial code and henceforth of a polar code."}]