Higher weight spectra of ternary codes associated to the quadratic Veronese $3$-fold
Krishna Kaipa, Puspendu Pradhan
TL;DR
This work determines the higher weight spectra of the ternary Projective Veronese C_3 code arising from the Veronese embedding of PG(3,q) into PG(9,q). The authors reduce the problem to finite-geometry counts of j-point configurations in PG(3,q) whose Veronese images span a given subspace in PG(9,q), and develop a two-step method: (i) classify maximal rank-r configurations \mathcal{C}_r and organize them into G-orbits, (ii) compute B_{j,r} by embedding counts within each orbit and assembling across orbits; this is complemented by using extended weight enumerators W(Z;T) and the Jurrius–Pellikaan framework to recover the r-th generalized weight enumerators. The method is implemented for q=3, yielding explicit generalized weights d_1=18, d_2=24, d_3=26, d_4=27, d_5=33, d_6=35, d_7=36, d_8=38, d_9=39, d_{10}=40 for C_3 over F_3, with a complete construction of the B_{j,i} polynomials up to rank 9. The results demonstrate a concrete pathway to higher-weight spectra for Veronese-associated codes and set the stage for general q in future work, highlighting the deep interplay between coding theory and finite geometry.
Abstract
The problem studied in this work is to determine the higher weight spectra of the Projective Reed-Muller codes associated to the Veronese $3$-fold $\mathcal V$ in $PG(9,q)$, which is the image of the quadratic Veronese embedding of $PG(3,q)$ in $PG(9,q)$. We reduce the problem to the following combinatorial problem in finite geometry: For each subset $S$ of $\mathcal V$, determine the dimension of the linear subspace of $PG(9,q)$ generated by $S$. We develop a systematic method to solve the latter problem. We implement the method for $q=3$, and use it to obtain the higher weight spectra of the associated code. The case of a general finite field $\mathbb F_q$ will be treated in a future work.