Minimal linear codes from Hermitian varieties and quadrics

In this note we investigate minimal linear codes arising from Hermitian varieties and quadrics. We study their parameters and formulate some open problems about their weight distribution.


Introduction
Let q be the finite field with q elements, and let C be a k-dimensional vectorial subspace of n q , where n ∈ ℕ . From now on, we will call C a [n, k] linear code endowed with the Hamming metric, i.e. the distance between two codewords (elements of C) is given by the number of the coordinate positions in which the two elements differ.
For a codeword c ∈ C , the support of c, denoted by Supp(c) , is the set of its nonzero coordinate positions, and the weight of c is wt(c) = #Supp(c) . A codeword c is said to be minimal if its support determines c up to a scalar multiplication, i.e. if c � ∈ C is linear independent with c, then Supp(c � )⊈Supp(c) . Minimal codewords were employed by Massey (see [10,11]) for the construction of a perfect and ideal 1 3 SSS, in which the access structure is determined by the set of the minimal codewords of a linear code.
Unfortunately, the determination of the set of minimal codewords of a given code is a difficult task. Therefore, obtaining the access structure of Massey's SSS is very challenging. This fact led to the study of codes for which every non-zero codeword is minimal, that are called minimal codes.
A useful tool to construct minimal codes was given by Ashikhmin and Barg (see [2]).

Lemma 1
Let Cbe a linear code over q , and denote by wt min and wt max the minimum and maximum nonzero weights in C , respectively. If wt min wt max > q−1 q (called the ABcondition from now on), then C is minimal.
Since the study of minimal codes has attracted much attention in recent years, many families of minimal linear codes satisfying AB have been constructed, e.g. see [5].
However, AB is only a sufficient condition and, for this reason, a relative new research line consists in finding examples of minimal codes violating AB . Families of these codes were first constructed in [7], whereas [6,8] give the first infinite family of minimal codes for the binary and ternary case, respectively. Afterwards, in [3] the first examples of minimal linear codes for every field of odd characteristic were constructed.
Following the geometrical approach of [3], in [4] it was proved that it is possible to construct families of minimal codes through the study of cutting blocking sets.
Recently, in [1,13] the authors found out independently that cutting blocking sets not only determine minimal codes, but they are actually in bijection with them.
Therefore, minimal codes and algebraic varieties over finite fields are very related objects. In this note we investigate families of minimal codes arising from celebrated objects in finite geometry, i.e. Hermitian varieties and quadrics, giving some information about their weight distribution.

Blocking sets
Hereafter, we will denote with q the finite field with q elements, where q = p h is a prime power, and with PG(n, q) the projective space of dimension n ∈ ℕ over q . A subspace of dimension r in PG(n, q) is denoted by r . For a set of homogeneous polynomials f 1 , … , f r ∈ q [X 0 , … , X n ] , V p (f 1 , … , f r ) is the projective variety associated to f 1 , … , f r . With r V p (f 1 , … , f r ) we denote a cone in PG(n, q) having vertex r and base V p (f 1 , … , f r ) , i.e. r V p (f 1 , … , f r ) is the join of r to V p (f 1 , … , f r ). In the following, we recall some useful basic definitions on blocking sets; for an exhaustive reference we point to [12,Chapter 3].
Definition 1 A k-blocking set is a subset of PG(n, q) intersecting all (n − k)-dimensional subspaces. When k = 1 , it is simply called a blocking set.
The dimension of a k-blocking set corresponds to the dimension of the subspace generated by its elements.
A (k, s)-blocking set in PG(n, q) , where n > k , is a k-blocking set that does not contain a s-dimensional subspace.
As we already mentioned, in order to construct minimal linear codes as in [4] we need to introduce a particular class of blocking set.

Definition 2
A k-blocking set in PG(n, q) is cutting if its intersection with every (n − k)-dimensional subspace is not contained in any other (n − k)-dimensional subspace.

Minimal codes from cutting blocking sets
For every homogeneous function f ∶ n+1 q → q , let C f be the linear code defined as Here we write a point of PG(n, q) in standard notation, i.e. the first nonzero coordinate from the left is 1. The resulting code does not depend on this choice.

Remark 1
If f is non-linear, then C f is a [(q n+1 − 1)∕(q − 1), n + 2] code and a generator matrix is obtained by extending the generator matrix of a simplex code by a line from the evaluation of the homogeneous function f.
In particular, for any The following result gives a sufficient condition to construct minimal codes (see [4] for details).
Theorem 1 Let f ∶ n+1 q → q be a homogeneous function such that: , n + 2] minimal code over q . Depending on the variety V p (f ) , the code C f may not satisfy the AB condition.
then the code C f as in (1) does not satisfy the AB condition. (1) Our aim is to show that properties (a) and (b) of Theorem 1 hold in the case of Hermitian varieties and quadrics.

Minimal codes from Hermitian varieties
Our notations and terminologies are standard, see [9]. Let f r , r ∈ {1, … , n} , be a canonical Hermitian form in The variety V p (f r ) is a Hermitian variety in PG(n, q 2 ) . If r = n , the Hermitian variety V p (f n ) is non-singular and it is denoted by H n .
The following result is a corollary of [9, Lemma 2.20] and gives information about the intersections of H n with hyperplanes. Propositions 1 and 2 allow to prove that Theorem 1 can be applied to non-singular Hermitian varieties.

Proposition 3 Condition (a) of Theorem 1 holds for non-singular Hermitian varieties.
Proof We show that H n is an n-dimensional (1, n − 1) cutting blocking set of PG(n, q) . First, by Proposition 1, H n is an n-dimensional blocking set. Moreover H n cannot contain a hyperplane (see Proposition 2), and hence it is a (1, n − 1) blocking set. Finally, since by Proposition 1 the intersection of H n with any hyperplane cannot be contained in a space of dimension n − 2 , the claim follows. (2) Remark 2 Since the number of points of a non-singular projective Hermitian variety of dimension n is Lemma 2 implies that C f does not satisfy the AB condition.

On the weight distribution of minimal codes from H n
In this section, A(H n ) = P(H n ) − P(H n−1 ) denotes the number of affine points of a non-singular Hermitian variety of PG(n, q 2 ) and ◻ * q is the set of nonzero squares of q . We collect here the weight of some codewords, as a first approach to the weight distribution problem. The next proposition is from Remark 1.
Proof Let A n be the number of (x 0 , … , x n ) ∈ n+1 q 2 such that the first non-zero coordinate is 1 and u(x n . For an (n + 1)-tuple satisfying X + Y = 0 , either X = 0 and Y = 0 or X ≠ 0 . In both cases X = −Y ∈ q . Let Ẽ j , j ≥ 1 , be the number of (j + 1) -tuples (1, On the other hand, the number of (i + 1)-tuple (x 0 , … , x i ) such that the first non-zero coordinate has value 1 and X ∈ q is q ⋅ q 2i −1 q 2 −1 + 1 . Hence the case X ≠ 0 and X + Y = 0 occurs in Assume now 1 − 4c ∈ ◻ * q . Then there exists w ∈ * q such that Since there are q−1 2 nonzero squares in q and −3 ∉ ◻ * q if and only if q ≡ 2 (mod 3) , the claim follows.
= 0 such that their first non-zero component is at least the (i + 1)-th one is P(H n−i−1 ) , the claim follows from Lemma 3. ◻

Minimal codes from quadrics
For a quadratic form f ∈ q [X 0 , … , X n ], with f = ∑ n i=0 a i X 2 i + ∑ i<j a i,j X i X j , the associated variety V p (f ) is a quadric in PG(n, q) . If f is non-degenerate, i.e. f cannot be reduced by a linear transformation to a quadratic form in less variables, V p (f ) is a non-singular quadric and it is denoted by Q n . Up to projective equivalence there are one or two distinct non-singular quadrics according to n being even or odd; see [9, Chapter 1].

Proposition 9
-If n is even a non-singular quadric Q n is projectively equivalent to a non-singular quadric P n , called parabolic, where

3
-If n is odd a non-singular quadric Q n is either projectively equivalent to a nonsingular quadric E n , called elliptic, or to a non-singular quadric H n , called hyperbolic, where with g(X 0 , X 1 ) = dX 2 0 + X 0 X 1 + X 2 1 , irreducible over q , d ∈ * q , and The quadrics P n , E n and H n as in Proposition 9 are said to be in canonical forms. We say that two non-singular quadrics have the same character if they are both parabolic (character 1), both hyperbolic (character 2) or both elliptic (character 0).

Proposition 10
Let n ≥ 3 . The intersection of a non-singular quadric Q n with an hyperplane is either a non-singular quadric Q n−1 of PG(n − 1, q) or a cone 0 Q n−2 , where Q n and Q n−2 have the same character.

Proposition 11
The maximum dimension of a subspace on a non-singular quadric Q n is either 1 2 (n − 2) , 1 2 (n − 1) , or 1 2 (n − 3) according to Q n being respectively parabolic, hyperbolic or elliptic.
This results allow us to prove properties (a) and (b) of Theorem 1 hold for nonsingular quadrics in canonical form.

Proposition 12
Let n ≥ 3.Then condition (a) of Theorem 1 holds for non-singular quadrics of PG(n, q).
Proof By Proposition 10, a non-singular quadric is a blocking set of PG(n, q) . Moreover, by Proposition 11, Q n cannot contain an hyperplane, whence it is a (1, n − 1) -blocking set. Finally, since the intersection of Q n with an hyperplane cannot be contained in a space of dimension n − 2 (see Proposition 10), the blocking set is also cutting. ◻

Proposition 13
Let Q n = V p (f )be a non-singular quadric in canonical form. Then Condition (b) of Theorem 1 holds.
. We prove the claim separately for elliptic, parabolic and hyperbolic quadrics.
Then either v 0 ≠ 0 or v 1 ≠ 0 . Assuming without loss of generalities v 0 ≠ 0 , then ◻ By Propositions 12 and 13, Theorem 1 applies and the codes C f arising from nonsingular quadrics in canonical form are [(q n − 1)∕(q − 1), n + 2] minimal codes over q . Note that in this case Lemma 2 does not apply, and indeed MAGMA evidences suggest that the AB condition holds.

Conclusion and open problems
In this paper we proved that Hermitian varieties and quadrics are cutting blocking sets, and hence they give place to minimal codes. It is worth noting that Theorem 1 and the construction provided in [1] yield two different families of minimal codes. However, while for the codes C f Lemma 2 allows to argue about the AB condition, for the minimal codes arising from [1] this is not possible. In the following we list some open problems: 1. Determine the full weight distribution of the minimal codes C f presented in this paper; 2. Study whether the minimal codes arising from Hermitian varieties and quadrics with the construction in [1] do not respect the AB condition. otherwise,