Sphere Packings Constructed from Codes

Abstract

Let F be a finite field. A code $\aleph$ over F of length n is simply a subset of Fⁿ. A typical point, or codeword, of $\aleph$ has the form u = (u₁, u2, ..., u _n ), where u _i ∈ F. For convenience, we say that a codeword or point is of type [µ^k∣υ^l∣⋯] if µ = ∣u _i ∣ for k choices of u _i , υ = ∣u _i for [itl} choices of u _i , etc. The Hamming distance between two codewords u and v of $\aleph$ is the number of coordinates at which they differ, and is denoted by ‖u, v‖ _H . The weight of a codeword u, w(u), is the number of its nonzero coordinates. Thus, ‖u, v‖ _H = w(u − v).