On the Classification of Certain Ternary Codes of Length 12

A ternary [n, k] code C is a k-dimensional vector subspace of Fn3 , where F3 denotes the finite field of order 3. The weight wt(x) of a vector x is the number of non-zero components of x. The minimum non-zero weight of all codewords in C is called the minimum weight of C. A ternary [n, k, d] code is a ternary [n, k] code with minimum weight d. Throughout this note, we denote the minimum weight of a code C by d(C). Shimada and Zhang [9] studied the existence of polarizations on the supersingular K3 surfaces in characteristic 3 with Artin invariant 1 (see [9, Theorem 1.5] for the details). This was done by reducing the problem of the existence of the polarizations to a problem of the existence of ternary [12, 5] codes C satisfying the following conditions: