# What does “conjugation of coordinates” mean with respect to GF(4) (quantum) codes

In On the classification of all self-dual additive codes over $\textrm{GF}(4)$ of length up to 12 by Danielsen and Parker, they state:

Two self-dual additive codes over $\textrm{GF}(4)$, $C$ and $C^\prime$, are equivalent if and only if the codewords of $C$ can be mapped onto the codewords of $C^\prime$ by a map that preserves self-duality. Such a map must consist of a permutation of coordinates (columns of the generator matrix), followed by multiplication of coordinates by nonzero elements from $\textrm{GF}(4)$, followed by possible conjugation of coordinates.

with the previous definition

Conjugation of $x \in \textrm{GF}(4)$ is defined by $\bar{x} = x^2$.

I am confused what "conjugation of coordinates" means in this context. To me "coordinates" would normally refers to the code matrix's columns, or equivalently the quantum code's qubits. However, here it seems to be referring to the alphabet of the code, or equivalently the Pauli operators of the code's stabilizer generators. If this is the case, what operation does "conjugation of coordinates" represent with respect to the code's stabilizer generators?

I am confused what "conjugation of coordinates" means in this context.

Conjugating coordinates of $\mathcal C$ is equivalent to setting some diagonal elements of Γ to 1.

Read "Theorem 12, on page 8 and 9" for an understanding of the usage, this is further explained on page 15 (last paragraph):

"As mentioned before, the set of self-dual linear codes over GF(4) is a subset of the self-dual additive codes of Type II. Note that conjugation of single coordinates does not preserve the linearity of a code. It was shown by Van den Nest $^{[25]}$ that the code $\mathcal C$ generated by a matrix of the form Γ + $ωI$ can not be linear. However, if there is a linear code equivalent to $\mathcal C$, it can be found by conjugating some coordinates. Conjugating coordinates of $\mathcal C$ is equivalent to setting some diagonal elements of Γ to 1. Let $A$ be a binary diagonal matrix such that Γ + $A$ + $ωI$ generates a linear code. Van den Nest $^{[25]}$ proved that $\mathcal C$ is equivalent to a linear code if and only if there exists such a matrix $A$ that satisfies Γ$^2$ + $A$Γ + Γ$A$ + Γ + $I$ = $0$. A similar result was found by Glynn et al. $^{[12]}$. Using this method, it is easy to check whether the LC orbit of a given graph corresponds to a linear code. However, self-dual linear codes over GF(4) have already been classified up to length 16, and we have not found a way to extend this result using the graph approach.".

References: [12] D. G. Glynn, T. A. Gulliver, J. G. Maks, M. K. Gupta, The geometry of additive quantum codes, submitted to Springer-Verlag, 2004. Book  [25] M. Van den Nest, Local Equivalence of Stabilizer States and Codes, Ph.D. thesis, K. U. Leuven, Leuven, Belgium, May 2005. .PDF (English starts on page 22)

• A minor nitpick. This answer is rather incomprehensible unless you go back and look up the definition of $\Gamma$ in the original paper, since $\Gamma$ doesn't appear in the OP. And I would think that "conjugating coordinates" should be a reversible operation, while "setting some diagonal elements of $\Gamma$ to $1$" isn't. – Peter Shor Jun 13 '18 at 14:29