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Nearest-neighbor error correcting codes on a hexagonal signal constellation

We propose a new class of single error correcting linear codes suitable for a two dimensional hexagonal constellation. The proposed code is a linear subspace of ℤ<sub>6n+1</sub><sup>n</sup> where n is code length and 6n + 1 is a prime number. It corrects a single error in the set |±1, ±α<sup>n</sup>, ±α<sup>2n</sup>} where α is a primitive element of ℤ<sub>6n+1</sub><sup>x</sup>. Moreover, we apply the proposed code to a two dimensional hexagonal constellation and show that it corrects an error such that a transmitted symbol moves to one of its nearest neighbors over the hexagonal constellation at the decoder side. We also consider an extension of the proposed code to double nearest neighbor error correcting codes. Some examples of such codes obtained by computer-assisted search are presented.