On the Closed-Form Weight Enumeration of Polar Codes: 1.5d -Weight Codewords

The weight distribution of an error correction code is a critical determinant of its error-correcting performance. In the case of polar codes, the minimum weight $\mathop {\mathrm {w}}\nolimits _{\min }$ (equal to the minimum distance d) is the only weight for which an explicit enumerator formula is currently available. Having closed-form weight enumerators for polar codewords with weights greater than the minimum weight not only simplifies the enumeration process but also provides valuable insights towards constructing better polar-like codes. In this paper, we contribute towards understanding the algebraic structure underlying higher weights by analyzing Minkowski sums of orbits. Our approach builds upon the lower triangular affine (LTA) group of decreasing monomial codes. Specifically, we propose a closed-form expression for the enumeration of codewords with weight $1.5\mathop {\mathrm {w}}\nolimits _{\min }$ . The key insight for code design is that the enumeration of codewords with weight $\mathop {\mathrm {w}}\nolimits _{\min }$ and $1.5\mathop {\mathrm {w}}\nolimits _{\min }$ relies on the set of maximum degree monomials. This set corresponds to the indices of minimum weight rows of the polar transform $\boldsymbol {G}_{N}$ belonging to the information set $\mathcal {I}$ . Consequently, reducing the cardinality of this set can lead to a reduction of the number of codewords in both weight categories.