Minimal Linear Codes Constructed from Sunflowers

Sunflower in coding theory is a class of important subspace codes and can be used to construct linear codes. In this paper, we study the minimality of linear codes over Fq constructed from sunflowers of size s in all cases. For any sunflower, the corresponding linear code is minimal if s≥q+1, and not minimal if 2≤s≤3≤q. In the case where 3<s≤q, for some sunflowers, the corresponding linear codes are minimal, whereas for some other sunflowers, the corresponding linear codes are not minimal.


Introduction
Let F q be the finite field with q elements and F n q the vector space with dimension n over F q .For a vector v = (v 1 , . . ., v n ) ∈ F n q , let Suppt(v) := {1 ≤ i ≤ n : v i ̸ = 0} be the support of v.The Hamming weight of v is wt(v):=#Suppt(v).For any two vectors u, v ∈ F n q , if Suppt(u) ⊆ Suppt(v), we say that v covers u (or u is covered by v) and write u ⪯ v. Clearly, av ⪯ v for all a ∈ F q .
An [n, m] q linear code C over F q is an m-dimensional subspace of F n q .A codeword c in a linear code C is called minimal if c covers only the codewords ac for all a ∈ F q , but no other codewords in C. If every codeword in C is minimal, then C is said to be a minimal linear code.Minimal linear codes have interesting applications in secret sharing [1][2][3][4][5] and secure two-party computation [6,7], and could be decoded with a minimum distance decoding method [8].
Up to now, there are two approaches to studying minimal linear codes.One is the algebraic method and the other is the geometric method.The algebraic method is based on the Hamming weights of the codewords.In [8], Ashikhmin and Barg gave a sufficient condition for a linear code to be minimal.Many minimal linear codes satisfying the condition w min w max > q−1 q are obtained from linear codes with few weights; for example [9,10].Cohen et al. [7] provided an example to show that the condition w min w max > q−1 q is not necessary for a linear code to be minimal.Ding, Heng, and Zhou [11,12] derived a sufficient and necessary condition on all Hamming weights for a given linear code to be minimal.
When using the algebraic method to prove the minimality of a given linear code, one needs to know all the Hamming weights in the code, which is very difficult in general.Even if all the Hamming weights are known, it is hard to use the algebraic method to prove the minimality.In this paper, we will use the geometric approaches to study the minimality of some linear codes.Based on the geometric approaches (see [13][14][15]) it is easier to construct minimal linear codes or to prove the minimality of some linear codes (see [16][17][18][19][20][21]).
Sunflower in coding theory is a class of important subspace codes and can be used to construct linear codes, see [22].Let s be the number of the elements in a sunflower.In [23], (Theorem 10), the authors proved that if s ≥ p + 1, then the corresponding linear code over F p is minimal, where p is a prime number.
In this paper, we will use the approach used in [14] to consider the minimality of linear codes over F q constructed from sunflowers for all s.We obtain the following three results: (1) when s ≥ q + 1, for any sunflower, the corresponding linear code is minimal; (2) when 2 ≤ s ≤ 3 ≤ q, for any sunflower, the corresponding linear code is not minimal; (3) when 3 < s ≤ q, for some sunflowers, the corresponding linear codes are minimal, wherea for some other sunflowers, the corresponding linear codes are not minimal.
This paper is organized as follows.In Section 2, we introduce some basic knowledge about sunflowers, Euclidean inner product, and minimal linear codes.In Section 3, we consider the linear codes constructed from sunflowers and discuss the minimality of these linear codes in three cases.In Section 4, we conclude this paper.

Sunflower
Throughout this paper, let k and t 0 be two positive integers, m = 2k + t 0 and l = k + t 0 .Let 2 ≤ s ≤ q k + 1 be a positive integer, T 0 ≤ F m q be a subspace of F m q , and dimT 0 = t 0 .We denote G q (l, m) the set of l-dimensional vector subspaces of F m q .We define Then, Φ ⊆ G q (l, m) is a sunflower of F m q and the space T 0 is called the center of the sunflower Φ.
Lemma 1.Let Φ ⊆ G q (l, m) be a sunflower and T 0 the center of Φ.For any E i , E j ∈ Φ with Lemma 2. Let Φ ⊆ G q (l, m) be a sunflower and T 0 the center of Φ.For any E i , E j ∈ Φ with , which implies z = 0.

Euclidean Inner Product
Let m be a positive integer.For x = (x 1 , x 2 , . . ., x m ), y = (y 1 , y 2 , . . ., y m ) ∈ F m q , the Euclidean inner product of x and y is given by For any S ⊆ F m q , we define S ⊥ := {v ∈ F m q | vs T = 0, for any s ∈ S}.

Minimal Linear Codes
All linear codes can be constructed by the following way.Let m ≤ n be two positive integers.Let G := [d 1 , . . ., d n ] be an m × n matrix over F q and D := {d 1 , . . ., d n } be a multiset.Let r(D) = r(G) denote the rank of G, which is equal to the dimension of the vector space Span(D) over Then, C(D) is an [n, r(D)] q linear code with generator matrix G.We always study the minimality of C(D) by considering some appropriate multisets D.
To present the sufficient and necessary condition for minimal linear codes in [14], some concepts are needed.For any y ∈ F m q , we define It is obvious that H(y, D) ⊆ V(y, D) ⊆ H(y).
Let y ∈ F m q \{0}.The following lemma gives a sufficient and necessary condition for the codeword c(y) ∈ C(D) to be minimal.

Lemma 3 ([14] (Theorem 3.1)).
Let y ∈ F m q \{0}.Then, the following three conditions are equivalent: The following lemma gives a sufficient and necessary condition for linear codes over F q to be minimal.Lemma 4 ([14] (Theorem 3.2)).The following three conditions are equivalent: By the following lemma, we can obtain infinity of many minimal linear codes from any known minimal linear codes.Lemma 5 ([14] (Proposition 4.1)).Let D 1 ⊆ D 2 be two multisets with elements in F m q and r(D The following corollary is trivial.
In the following section, we will use the above lemmas to consider the minimality of linear codes constructed from sunflowers.

The Minimality of Linear Codes Constructed from Sunflowers
In this section, we consider the linear codes constructed from sunflowers and discuss the minimality of these linear codes. Let be a sunflower of F m q and T 0 the center of Φ.Let It is easy to see that C(D) is a [s(q l − q t 0 ), m] q linear code.
The following lemmas are important in the proofs of this section.
Proof.Since y / ∈ E ⊥ i , it follows from Lemma 6 that dimH(y, Then, we have If y / ∈ T ⊥ 0 , then dimH(y, T 0 ) = t 0 − 1 by Lemma 6. Suppose that Then, we have The proof is completed.Now, we consider the minimality of C(D) in three cases.First, when s ≥ q + 1, we have Theorem 1.Let Φ = {E 1 , . . ., E s } be a sunflower of F m q with center T 0 of dimension t 0 .If s ≥ q + 1, then C(D) is an [s(q l − q t 0 ), m] q minimal linear code.
Proof.According to Lemma 4, we only need to prove that for any y ∈ F m q \{0}, dimV(y, D) = m − 1.By (2), we obtain There are three cases: (1) If there exists E i 0 ∈ Φ such that y ∈ E ⊥ i 0 , then we have dimH(y, E i 0 ) = l from Lemma 6.According to Lemma 2, for any E j 0 ∈ Φ with E j 0 ̸ = E i 0 , we have y / ∈ E ⊥ j 0 .Then, it follows from Lemma 6 that dimH(y, When k > 1, we set By (3), we have it is easy to obtain dimV(y, D) = m − 1.
(ii) If for any E i ∈ Φ we have dimπ(E i ) = 1, combining that V = E i + E j for any E i , E j ∈ Φ with E i ̸ = E j in accordance with Lemma 1, we have Since V has only q + 1 one-dimensional subspace and s ≥ q + 1, we have s = q + 1 and Hence, there exists α = α k − bβ k + w ∈ E j 0 , where w ∈ W, such that π(α) = α k − bβ k .One can easily deduce that α / ∈ T 0 , α ∈ H(y) and α / ∈ W. We obtain Thus, dimV((y), D) = m − 1.
In conclusion, for any y ∈ F m q \{0}, we have dimV(y, D) = m − 1, so C(D) is a minimal linear code.

Remark 1.
In Theorem 1, if q = p is a prime number, then it becomes [23] (Theorem 10).So Theorem 1 is a generalization of [23] (Theorem 10).Our method is different from theirs.When s ≤ q, our method also can be used to study the minimality of the linear codes, whereas theirs can not.
Example 1.Let e 1 , . . ., e m be the standard basis of F m q .Let For any b ∈ F q , we define E b = Span{e 1 + be k+1 , e 2 + be k+2 , . . ., e k + be 2k , e 2k+1 , . . ., e m }. (6) Suppose that It is easy to see that Φ is a sunflower of F m q with center T ′ 0 and s = q + 1.Here, we take q = 4, k = 3, and t 0 = 1.With the help of Magma, we verify that the code C(D ′ ) is a minimal [1260, 7] 4 linear code with minimum distance 768, and Now, we consider the minimality of C(D) when 2 ≤ s ≤ 3 ≤ q.If s = 3, we have Theorem 2. Let Φ = {E 1 , . . ., E s } be a sunflower of F m q with center T 0 of dimension t 0 .If s = 3 ≤ q, then C(D) is not minimal.
Proof.To prove C(D) is not minimal, by Lemma 4, we only need to prove there exists and dimH(y 1 , T 0 ) = t 0 .We set 1 to E ′ 2 and Since Thus, C(D) is not minimal. ( ).Then, dimV(y 0 , D) = m − 2. By Lemma 4, we have that c(y 0 ) is not minimal.
We will show that some sunflowers Φ with center T ′ 0 , C(D) are minimal, whereas some other sunflowers Φ with center T ′ 0 , C(D) are not minimal.First, we construct some sunflowers Φ such that C(D) are minimal.Let k ≥ 2, f (x) be an irreducible polynomial in F q [x] of degree k and M ∈ F k×k q be a matrix with characteristic polynomial f (x).We define and We can see Φ is a sunflower with center T ′ 0 .
Theorem 3.For the sunflower Φ defined in (8), the linear code C(D) is minimal.
Proof.According to Lemma 4, we only need to prove that for any y ∈ F m q \{0}, dimV(y, D) = m − 1.There are three cases: (1) If there exists E i 0 ∈ Φ such that y ∈ E ⊥ i 0 , the proof is similar as that in Theorem 1 (1).(2) If for any E i ∈ Φ, 1 ≤ i ≤ s, we have y / ∈ E ⊥ i and y / ∈ T ′⊥ 0 , then the proof is similar to that in Theorem 1 (2).
That is to say, H(y 1 ) is the ψ-invariant subspace of F k q .Let α 1 , . . ., α k−1 , α k be a basis of F k q , where α 1 , . . ., α k−1 is a basis of H(y 1 ).Then, the matrix of ψ with respect to this basis is where B 1 is the matrix of ψ|H(y 1 ) with respect to α 1 , . . ., α k−1 .Note that M is the matrix of ψ with respect to the standard basis, and thus M and B are similar and have the same characteristic polynomial.So Combining Theorem 3 and Lemma 5, we have Corollary 3. Let s ≥ 4 and Φ = {E 1 , . . ., E s } be a sunflower of F m q with center T ′ 0 .If {E 1 , E 2 , E 3 , E 4 } are defined as (7), then C(D) is minimal.
Example 2. Take q = 5, k = 2, and t 0 = 1.Let f (x) = x 2 + x + 1 and It is easily checked that f (x) ∈ F q [x] is an irreducible polynomial of degree 2 and the characteristic polynomial of M.Then, the code C(D) constructed based on Theorem 3 is a minimal [480, 5] 5 linear code with minimum distance 300, and Now, we construct some sunflowers Φ with center T ′ 0 such that C(D) are not minimal.Let us recall from ( 6) that E b = Span{e 1 + be k+1 , e 2 + be k+2 , . . ., e k + be 2k , e 2k+1 , . . ., e m }.
It is easy to see that Φ is a sunflower of F m q with center T ′ 0 .