Isometry-dual flags of AG codes

Abstract

Consider a complete flag $\{0\} = C_0< C_1< \cdots < C_n = \mathbb {F}^n$ of one-point AG codes of length n over the finite field $\mathbb {F}$. The codes are defined by evaluating functions with poles at a given point Q in points $P_1,\dots ,P_n$ distinct from Q. A flag has the isometry-dual property if the given flag and the corresponding dual flag are the same up to isometry. For several curves, including the projective line, Hermitian curves, Suzuki curves, Ree curves, and the Klein curve over the field of eight elements, the maximal flag, obtained by evaluation in all rational points different from the point Q, is self-dual. More generally, we ask whether a flag obtained by evaluation in a proper subset of rational points is isometry-dual. In Geil et al. (2011) it is shown, for a curve of genus g, that a flag of one-point AG codes defined with a subset of $n > 2g+2$ rational points is isometry-dual if and only if the last code $C_n$ in the flag is defined with functions of pole order at most $n+2g-1$. Using a different approach, we extend this characterization to all subsets of size $n \ge 2g+2$. Moreover we show that this is best possible by giving examples of isometry-dual flags with $n=2g+1$ such that $C_n$ is generated by functions of pole order at most $n+2g-2$. We also prove a necessary condition, formulated in terms of maximum sparse ideals of the Weierstrass semigroup of Q, under which a flag of punctured one-point AG codes inherits the isometry-dual property from the original unpunctured flag.

Appendix: Reed Muller type code

Since the isometry-dual property of a complete flag of codes can be defined with respect to the generator matrix, we consider an example of codes not necessarily defined over a curve. Consider the affine space $\mathbb {F}_2^m$. It is an m dimensional vector space over $\mathbb {F}_2$, so elements are of the form $(\alpha _m, \alpha _{m-1}, \ldots , \alpha _1)$ where each $\alpha _j \in \mathbb {F}_2 = \{0,1\}$ for $j=1, \ldots , m$. Let $x_{\alpha _j}$ be the coordinate functions for $j=1, \ldots , m$, that is, $x_{\alpha _j} ( \alpha _m, \alpha _{m-1}, \ldots , \alpha _1) = \alpha _j$ for $j=1, \ldots , m$. Note that $x_{\alpha _j}^2 = x_{\alpha _j}$. Then the coordinate ring is $R=\mathbb {F}_2 [ x_1, x_2, \ldots , x_m]/I$ for $I=(x_1^2 - x_1, x_2^2 - x_2, \ldots , x_m^2 - x_m)$, which is, as a set, a set of square free monomials in $x_1, \ldots , x_m$. Let $\alpha =(\alpha _m, \alpha _{m-1}, \ldots , \alpha _1)$ and $\beta =(\beta _m, \beta _{m-1}, \ldots , \beta _1)$ be vectors in $\mathbb {F}_2^m$. We write $x^\alpha$ for the function $x_1^{\alpha _1} x_2^{\alpha _2}\cdots x_m^{\alpha _m}$. An order on R is defined by $x^\alpha < x^\beta$ if and only if

• 1. Either $\sum \alpha _i < \sum \beta _i$

• 2. or $\sum \alpha _i = \sum \beta _i$ and $\exists j$ such that $\alpha _j =0$, $\beta _j=1$ and $\alpha _k=\beta _k$ for all $k=j+1, \ldots , m$.

Call this by DegLex order. There is a bijection between $\mathbb {F}_2$-rational points of $\mathbb {F}_2^m$ and functions in the coordinate ring R by $\alpha \longleftrightarrow x^\alpha$. We copy the DegLex order on R to $\mathbb {F}_2^m$. For a function $f=X^\alpha$ and a point $P = \beta$,

$$\begin{aligned} f(P) = {\left\{ \begin{array}{ll} 1 \;\;\text { if } x^\alpha | x^\beta \\ 0 \;\;\text { otherwise} \end{array}\right. } \end{aligned}$$

Let $N=2^m$. Define an $N\times N$ matrix $A=(A_{f,P})$ with rows of functions in R and columns of affine points in $\mathbb {F}_2^m$ both indexed by DegLex order. Then with this orders on points and functions, we get an isometry-dual matrix. For $m=3$, we get the following matrix A.

	000	001	010	100	011	101	110	111
1	1	1	1	1	1	1	1	1
$x_1$	0	1	0	0	1	1	0	1
$x_2$	0	0	1	0	1	0	1	1
$x_3$	0	0	0	1	0	1	1	1
$x_1 x_2$	0	0	0	0	1	0	0	1
$x_1 x_3$	0	0	0	0	0	1	0	1
$x_2 x_3$	0	0	0	0	0	0	1	1
$x_1 x_2 x_3$	0	0	0	0	0	0	0	1

Then it can be easily checked that this matrix is isometry-dual with the dualizing vector $(1, 1, \ldots , 1)$, that is, we get

$$\begin{aligned} A A^T = \left[ \begin{array}{cccccccc} 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 1 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 1 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 1 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 1 &{}\quad 1 &{}\quad 1 \\ 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 1 &{}\quad 1 \\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 1 \\ 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 \\ \end{array} \right] \end{aligned}$$

For a subset of n rational points, the corresponding columns in A define a $N \times n$ submatrix whose row spaces define a flag of length n from 0 to $\mathbb {F}_2^n$. From this $N\times n$ matrix choose n rows in a way that the chosen row is linearly independent on the previous rows. Then we get a complete flag of $\mathbb {F}_2^n$ generated by the first i rows of the $n\times n$ matrix. We are interested in the case when this choice gives an isometry-dual flag. The next two tables give the relevant minors that define the flag for the subsets of points $\{ 000, 001, 010, 011 \}$ and $\{ 000, 001, 010, 111 \}$.

	000	001	010	011		000	001	010	111
1	1	1	1	1	1	1	1	1	1
$x_1$	0	1	0	1	$x_1$	0	1	0	1
$x_2$	0	0	1	1	$x_2$	0	0	1	1
$x_1 x_2$	0	0	0	1	$x_3$	0	0	0	1

The two subsets share the same minors and thus the same flags. Both are isometry-dual. For the first subset this follows immediately with the observation that the set is the set of all rational points in affine space of dimension 2, that is, it is in the affine plane which is a hyperplane of a coordinate function $x_3$. The vanishing ideals for the two sets of points are

$$\begin{aligned} I( \{ 000, 001, 010, 011 \} )&= (x_3), \qquad \\ I( \{ 000, 001, 010, 111 \} )&= (x_1 x_2 + x_3, x_1 x_3 + x_3, x_2 x_3 + x_3). \end{aligned}$$

There are a total of 22 isometry-dual subsets of size 4. The row span $R_5$ of the first five rows in the 8-by-8 matrix A, the rows labeled $1, x_1, x_2, x_3 x_1 x_2$, contains 32 vectors, with weight distribution $0\,(1\times ), 2\,(4\times ), 4\,(22\times ), 6\,(4\times ), 8\,(1\times ).$ The 22 vectors of weight 4 are the characteristic vectors of the 22 isometry-dual subsets of size 4. They are divided into four groups: 2 are in $R_2 \backslash R_1$, 4 are in $R_3 \backslash R_2$, 8 are in $R_4 \backslash R_3$, and 8 are in $R_5 \backslash R_4$. Minors for subsets in the same group share the same rows.

$$\begin{aligned}&R_2 \backslash R_1 : 1, x_2, x_3, x_2 x_3~(2\times ) \\&R_3 \backslash R_2 : 1, x_1, x_3, x_1 x_3~(4\times ) \\&R_4 \backslash R_3 : 1, x_1, x_2, x_1 x_2~(8\times ) \\&R_5 \backslash R_4 : 1, x_1, x_2, x_3~(8\times ) \end{aligned}$$

Let $v=(1,1,1,0,0,0,0,1)$ be the characteristic vector for the subset $\{ 000, 001, 010, 111 \}$.

$$\begin{aligned} A \hbox {diag}(v) A^T = \left[ \begin{array}{cccccccc} 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 \\ 0 &{}\quad 0 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 \\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 \\ 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 \\ 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 \\ 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 \\ 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 \\ 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 \end{array} \right] \end{aligned}$$

The next propositions are proved computationally.

Proposition 26

There are $54 = 2^6-10$ isometry-dual subsets of size 8 in the affine space $\mathbb {F}_2^4$. Their characteristic vectors are the vectors of weight 8 in the row span $R_6$ of rows corresponding to $1, x_1, x_2, x_3, x_4, x_1 x_2$ in A. The weight distribution of the row span is $0^1 4^4 8^{54} 12^4 16^1$. The 54 subsets are divided into 6 orbits of sizes 2, 4, 8, 16, 8, 16. Pivots are in positions $(x^\alpha ,x^{\alpha '})$ :

$$\begin{aligned} x^\alpha x^{\alpha '} = {\left\{ \begin{array}{ll} x_2 x_3 x_4 ~~(2\,\times ) \\ x_1 x_3 x_4 ~~(4\,\times ) \\ x_1 x_2 x_4 ~~(8\,\times ) \\ x_1 x_2 x_3 ~~(16\,\times ) \\ x_3 x_4 ~\text {or}~ x_1 x_2 x_4~~(8\,\times ) \\ x_3 x_4 ~\text {or}~ x_1 x_2 x_3~~(16\,\times ) \end{array}\right. } \end{aligned}$$

Ideals that represent the different groups are

$$\begin{aligned} I = {\left\{ \begin{array}{ll} (x_1) \\ (x_2) \\ (x_3) \\ (x_4) \\ (x_3 + x_1 x_2,\, x_3 + x_1 x_3,\, x_3 + x_2 x_3) \\ (x_4 + x_1 x_2,\, x_4 + x_1 x_4,\, x_4 + x_2 x_4) \end{array}\right. } \end{aligned}$$

Proposition 27

There are $118 = 2^7-10$ isometry-dual subsets of size 16 in affine space $\mathbb {F}_2^5$. Their characteristic vectors are the vectors of weight 16 in the row span $R_7$ of rows corresponding to $1, x_1, x_2, x_3, x_4, x_5, x_1 x_2$ in A. The weight distribution of the row span is $0^1 8^4 16^{118} 24^4 32^1$. The 118 subsets are divided into 8 orbits of sizes 2, 4, 8, 16, 32, 8, 16, 32.