Cyclotomic Fields in Finite Geometry 2025, 30 - Cyclic Codes

A linear [n, k] code over 𝔽_q is a subspace C with dim k. Assuming (q, n) = 1, the elements of C are "codewords", with a weight determined by the number of its nonzero entries. The weight distribution of C is W: ℤ₀⁺ → ℤ₀⁺ , W(i) being the number of codewords in C of weight i. If W takes up to two different nonzero values, it's a two-weight code. C is cyclic iff it's an ideal in 𝔽_q[x]/(xⁿ - 1) =: F. A cyclic code C in F with generator polynomial g, if g = (xⁿ - 1)/f with f an irreducible divisor of xⁿ - 1, then C is an irreducible cyclic order over 𝔽_q (minimal cyclic code). For q prime, and n a positive integer with (n, q) = 1, then k := ord_n(q), β a primitive nth root of unity in 𝔽_qk. C_β = {c(y) := (Tr(y), Tr(yβ),..., Tr(yβ^n-1))^t: y ∈ 𝔽_q^k} is irreducible cyclic. Such can be reduced to those where n is divisible by q - 1. C_β is a 2-weight code iff ⟨β⟩ is a projective 2-intersection set in PG(k-1, q).

Define the sub-difference set of D in G/N D' := {Ng: g∈G, | D∩Ng | = a} for N a normal subgroup of G, and D having at least 2 different intersection numbers a, b with cosets of N in G. C_β is a 2-weight code iff the Singer difference set D = {d𝔽*_q: d ∈ 𝔽*_q^k, Tr(d) = 0} of PG(k-1, q) has a sub-difference set in G/⟨β⟩ with G = 𝔽*_q^k/𝔽*_q is the Singer cycle of PG(k-1, q).