In the work [1] it was shown that stabilizer codes could be mapped to binary codes (albeit with a symplectic norm) which further can be mapped to additive codes over GF(4). Can someone explain why it is useful to work over GF(4) rather than binary codes ?

[1] : Quantum Error Correction Via Codes Over GF(4), Robert Calderbank, Eric M. Rains, P. W. Shor, and Neil J. A. Sloane https://arxiv.org/pdf/quant-ph/9608006.pdf


1 Answer 1


Stabiliser codes map to linear subspaces of $\mathbb F_2^{2n}$ which are isotropic w.r.t. the standard symplectic form on this vector space. This is very useful for analysing these codes directly and explicitly computing plenty of properties.

However, it does not align very well with classical coding theory which deals with Euclidean spaces. Computer scientists had already become interested in codes over other finite fields, in particular $\mathbb F_4 = GF(4)$. Thus it seems like a good idea to map said isotropic subspaces over $\mathbb F_2$ to additive self-orthogonal codes over $\mathbb F_4$. If you consider the background of the authors, you might understand why this felt like a natural mapping to them.

And indeed, considering the papers of Calderbank, Rains, Shor, Sloane, Nebe, ... around and after 2000, they have thrown a lot of classical coding theory on the problem to establish some fundamental theorems on stabiliser codes (e.g. http://arxiv.org/abs/math/0001038). The papers, however, are quiet mathematical so a lot of implications might not be immediately clear.

Nevertheless, stabiliser codes are more than just classical additive self-orthogonal codes over $\mathbb F_4$ since this only reflects the structure of the stabiliser group up to phases. Often that's enough but not always.

  • $\begingroup$ Out of curiosity, what is the fundamental theorem on stabilizer codes that you are referring to in arxiv.org/abs/math/0001038 ? $\endgroup$ Commented Aug 9, 2022 at 15:37
  • $\begingroup$ @IanGershonTeixeira That paper is more on the automorphisms and the Clifford group. Admittedly, the reference is somewhat misplaced here. Take for instance arxiv.org/abs/quant-ph/9610040 . $\endgroup$ Commented Aug 10, 2022 at 7:28

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