On the l-extendability of quaternary linear codes

Finite Fields and Their Applications 35 (2015) 159–171

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Finite Fields and Their Applications www.elsevier.com/locate/ffa

On the l-extendability of quaternary linear codes H. Kanda, T. Tanaka, T. Maruta ∗,1 Department of Mathematics and Information Sciences, Osaka Prefecture University, Sakai, Osaka 599-8531, Japan

a r t i c l e

i n f o

Article history: Received 10 February 2014 Received in revised form 19 February 2015 Accepted 23 April 2015 Available online 17 May 2015 Communicated by James W.P. Hirschfeld

a b s t r a c t An [n, k, d]q code C is l-extendable if C can be extended to an [n + l, k, d + l]q code. We give some new sufficient conditions for the l-extendability of [n, k, d]4 codes with d ≡ 0 (mod 4) using the known results about odd sets in PG(k − 1, 4). The 3-extendability of quaternary linear codes is investigated through their diversities for the first time. © 2015 Elsevier Inc. All rights reserved.

MSC: 51E20 94B27 Keywords: Linear codes Extension Finite projective spaces Odd sets

1. Introduction Let Fq denote the field of q elements. An [n, k, d]q code is a linear code over Fq of length n with dimension k and minimum Hamming distance d. The weight distribution * Corresponding author. E-mail addresses: [email protected] (H. Kanda), [email protected] (T. Tanaka), [email protected] (T. Maruta). 1 The research of this author is partially supported by Grant-in-Aid for Scientific Research of Japan Society for the Promotion of Science under Contract Number 24540138. http://dx.doi.org/10.1016/j.ffa.2015.04.004 1071-5797/© 2015 Elsevier Inc. All rights reserved.

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of C is the list of numbers Ai which is the number of codewords of C with weight i. The weight distribution with (A0 , Ad , . . .) = (1, α, . . .) is also expressed as 01 dα · · ·. The diversity of C is defined from the weight distribution as the pair (Φ0 , Φ1 ) with Φ0 =

 1 Ai , q−1

Φ1 =

q|i,i>0

1 q−1



Ai .

i≡0,d (mod q)

For an [n, k, d]q code C with generator matrix G, C is l-extendable if there exist l vectors T h1 , . . . , hl ∈ Fkq such that the extended matrix [G, hT 1 , . . . , hl ] generates an [n +l, k, d +l]q code C  , and C  is an l-extension of C. Especially when l = 1, C is called extendable and C  is an extension of C. We can often see the extendability of a given code from its diversity. Such an approach was first given for ternary linear codes in [7]. In this paper, we deal with the extendability of quaternary linear codes. Extension theorems are employed to find optimal linear codes to construct new codes from old ones or to prove the nonexistence of codes with certain parameters; see [2] and [10]. In this paper, we consider [n, k, d]4 codes with d ≡ 0 (mod 4). For 1 ≤ j ≤ 3, we set Φ1,j =

1 3



Ai .

i≡j (mod 4)

Note that Φ1 = Φ1,3 + Φ1,2 if d ≡ 1 (mod 4); Φ1 = Φ1,1 + Φ1,3 if d ≡ 2 (mod 4); Φ1 = Φ1,1 + Φ1,2 if d ≡ 3 (mod 4). Denote by θj the number of points in PG(j, 4), i.e., θj = (4j+1 − 1)/3. We set θ0 = 1 and θj = 0 for j < 0 for convenience. The following is known for [n, k, d]4 codes with d odd. Theorem 1.1. (See [6,8,11].) Let C be an [n, k, d]4 code with diversity (Φ0 , Φ1 ), k ≥ 3, d odd. Then C is extendable if one of the following conditions holds: (a) (b) (c) (d) (e)

Φ0 = θk−4 , Φ1 = Φ1,2 , Φ1,2 = 0, Φ0 + Φ1,2 < θk−2 + 4k−2 , Φ0 + Φ1,2 = θk−2 + 2 · 4k−2 .

As for [n, k, d]4 codes with d ≡ 2 (mod 4), the following results are also known. Theorem 1.2. (See [14].) Let C be an [n, k, d]4 code with diversity (Φ0 , Φ1 ), k ≥ 3, d ≡ 2 (mod 4). Then C is extendable if Φ1,1 = 0 and Φ1,3 > 0. Theorem 1.3. (See [13].) Let C be an [n, k, d]4 code with diversity (Φ0 , 0), k ≥ 3, d ≡ 2 (mod 4). Then C is 2-extendable if Φ0 < θk−2 + 2 · 4k−3 or Φ0 > θk−2 + 2 · 4k−2 − 4. Moreover, for a generator matrix G of C, a 2-extension of C can be obtained by adding some column vector twice to G.

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161

We show that the condition (e) in Theorem 1.1 can be weakened to the following. Theorem 1.4. Let C be an [n, k, d]4 code with diversity (Φ0 , Φ1 ), k ≥ 3, d odd. Then C is extendable if Φ0 + Φ1,2 > θk−2 + 2 · 4k−2 − 4. The aim of this paper is to give some new sufficient conditions for the l-extendability of [n, k, d]4 codes with d ≡ 4 − l (mod 4), 1 ≤ l ≤ 3. The 3-extendability of quaternary linear codes is investigated through their diversities for the first time. The following theorems are our main results. Theorem 1.5. Let C be an [n, k, d]4 code with diversity (Φ0 , Φ1 ), k ≥ 3, d ≡ 1 (mod 4). Then C is 3-extendable if one of the following conditions holds: (1) Φ0 = θk−4 , (2) Φ1,2 = 0, (3) Φ1,3 = 0. Theorem 1.6. Let C be an [n, k, d]4 code with diversity (Φ0 , 4k−1 ), k ≥ 3, d ≡ 2 (mod 4). Then C is 2-extendable if Φ0 < θk−3 + 2 · 4k−4 or Φ0 > θk−3 + 2 · 4k−3 − 4. Theorem 1.7. Let C be an [n, k, d]4 code with diversity (θk−2 + 4k−2 , 2 · 4k−2 ), k ≥ 3, d ≡ 2 (mod 4). Then C is 2-extendable if Φ1,3 = 0. Theorem 1.8. Let C be an [n, k, d]4 code with diversity (Φ0 , Φ1 ), k ≥ 3, d ≡ 2 (mod 4). Then C is 2-extendable if one of the following conditions holds: (1) Φ1,1 = 0 and Φ1,3 > 0, (2) Φ1,1 > 0 and Φ1,3 = 0. Theorem 1.9. Let C be an [n, k, d]4 code with diversity (Φ0 , Φ1 ), k ≥ 4, d ≡ 0 (mod 4). If θk−4 < Φ0 ≤ θk−3 , then (1) (2) (3) (4)

Φ0 = θk−3 . C is extendable. C is 2-extendable when d is even. C is 3-extendable when d ≡ 1 (mod 4) and Φ1,2 = 3 · 4k−2 .

Theorem 1.10. Let C be an [n, k, d]4 code with diversity (Φ0 , Φ1 ), k ≥ 4, d odd. If C is not extendable, then Φ0 ≥ θk−3 + 2. We give some examples of quaternary linear codes below to which our results can be applied. Let F4 = {0, 1, ω, ω ¯ }, where ω and ω ¯ are the roots of x2 + x + 1 ∈ F2 [x]. We denote ω and ω ¯ by 2 and 3, respectively, for simplicity.

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Example 1.11. Let C1 be the [27, 5, 17]4 code with generator matrix ⎡

3 ⎢1 ⎢ ⎢ G1 = ⎢ 1 ⎢ ⎣1 3

3 1 1 3 1

3 1 3 1 1

3 2 2 3 3

3 2 3 2 3

3 2 3 3 2

3 3 1 1 1

3 3 2 2 3

3 3 2 3 2

3 3 3 2 2

3 1 1 1 1

3 2 3 3 3

3 3 2 3 3

3 3 3 2 3

3 3 3 3 2

3 1 1 2 2

3 1 2 2 3

3 1 2 3 1

3 1 3 1 2

3 2 1 1 3

3 2 1 3 2

3 2 2 1 1

3 2 3 2 1

3 3 1 2 1

3 3 2 1 2

0 0 0 3 1

⎤ 0 3⎥ ⎥ ⎥ 1⎥. ⎥ 0⎦ 0

Then, C1 has weight distribution 01 1760 18219 19246 2015 2181 22201 23183 253 2612 273 with diversity (5, 288), and C1 is 3-extendable by Theorem 1.5. We get a [30, 5, 20]4 code with weight distribution 01 20378 22162 24339 26126 2818 by adding the columns (3, 0, 2, 1, 0)T , (2, 3, 0, 0, 1)T and (3, 2, 1, 2, 1)T to G1 . Example 1.12. Let C2 be the [25, 4, 10]4 code with generator matrix ⎡

1 ⎢0 ⎢ G2 = ⎢ ⎣0 0

0 1 0 0

0 0 1 0

0 0 0 1

1 1 1 1

1 0 0 1

1 0 0 2

1 1 0 0

1 1 0 0

1 2 0 0

1 2 0 0

1 3 0 3

1 3 1 0

1 0 3 3

0 1 2 2

0 1 1 0

0 1 1 0

0 1 3 1

0 1 0 3

0 0 1 1

0 0 1 3

1 2 1 3

1 0 1 2

1 0 1 3

⎤ 1 1⎥ ⎥ ⎥. 0⎦ 1

Then, C2 has weight distribution 01 103 133 1512 163 1745 1818 1984 2030 2142 229 236 with diversity (11, 64), and C2 is 2-extendable by Theorem 1.6. Adding the columns (1, 3, 0, 0)T and (0, 1, 2, 1)T to G2 , we get a [27, 4, 12]4 code with weight distribution 01 123 143 1612 173 1815 1930 2063 2151 2248 2312 2415 . Example 1.13. Let C3 be the [16, 4, 7]4 code with generator matrix ⎡

1 ⎢0 ⎢ G3 = ⎢ ⎣0 0

0 1 0 0

0 0 1 0

0 0 0 1

1 1 1 1

0 0 1 0

1 1 0 2

1 1 0 2

1 1 2 3

0 2 1 2

1 2 1 1

1 0 2 2

1 1 0 0

1 1 2 2

1 2 3 3

⎤ 1 2⎥ ⎥ ⎥. 1⎦ 1

Then, C3 has weight distribution 01 76 83 912 1018 1184 1212 1360 1430 1530 with diversity (5, 40), and C3 is extendable by Theorem 1.9. We get a [17, 4, 8]4 code with weight distribution 01 89 1030 1296 1490 1630 by adding the column (1, 2, 0, 1)T to G3 . In the next section, we give the geometric method to investigate the l-extendability of codes over Fq . We prove Theorems 1.4–1.8 and 1.10 in Section 2 and Theorem 1.9 in Section 3. 2. Geometric approach We denote by PG(r, q) the projective geometry of dimension r over Fq . For an integer k ≥ 3, let Σ = PG(k − 1, q). A j-flat is a projective subspace of dimension j in Σ.

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The 0-flats, 1-flats, 2-flats, 3-flats, (k − 3)-flats and (k − 2)-flats in Σ are called points, lines, planes, solids, secundum and hyperplanes, respectively. We refer to [3] and [4] for geometric terminologies. For j < 0, a j-flat is the empty set as the usual convention. We investigate linear codes over Fq through projective geometry. Let C be an [n, k, d]q code with a generator matrix G and let gi be the i-th row of G (1 ≤ i ≤ k). For P = P(p1 , . . . , pk ) ∈ Σ, the weight of P with respect to G, denoted by wG (P ), is defined as k  wG (P ) = wt( pi gi ). i=1

Let Fd = {P ∈ Σ | wG (P ) = d}. Recall that a hyperplane H of Σ is defined by a non-zero vector h = (h1 , . . . , hk ) ∈ Fkq as H = {P(p1 , . . . , pk ) ∈ Σ | h1 p1 + · · · + hk pk = 0}. h is called the defining vector of H. Lemma 2.1. (See [9].) An [n, k, d]q code C is extendable if and only if there exists a hyperplane H of Σ such that Fd ∩ H = ∅. Moreover, the extended matrix of G by adding the defining vector of H as a column generates an extension of C. Now, let C be an [n, k, d]q code with d ≡ 0 (mod q) and let FC (i) = {P ∈ Σ | wG (P ) ≡ i (mod q)}, F1 = {P ∈ Σ | wG (P ) ≡ 0, d (mod q)}, F = Σ \ FC (d). Then we have Φ0 = |FC (0)|, Φ1 = |F1 |. Note that Fd ∩ FC (0) = ∅, Fd ⊂ FC (d). As a corollary of Lemma 2.1, we get the following. Corollary 2.2. C is extendable if there exists a hyperplane H of Σ such that H ⊂ F . We consider the extendability of C from this geometrical point of view. The following three lemmas give sufficient conditions for the l-extendability of codes over Fq , which will be used in the proofs of our main results. Lemma 2.3. Let C be an [n, k, d]q code with generator matrix G, k ≥ 3. If

d+l≤i
FC (i)

contains a hyperplane H of Σ, where 1 ≤ l < q, then C is l-extendable and an l-extension of C can be obtained by adding the defining vector of H repeated l times to G.

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Proof. We proceed by induction on l. The lemma follows from Lemma 2.1 for l = 1. Assume our assertion holds for l − 1 with l ≥ 2. Let H be a hyperplane of Σ contained in FC (i) and let C  be the extension of C with generator matrix G = [G, hT ], d+l≤i
where h is the defining vector of H. Since wG (P ) = wG (P ) for P ∈ H and wG (P ) = wG (P ) + 1 for P ∈ / H, we obtain

H⊂



FC (i) ⊂

d+l≤i
FC  (i).

d+1+(l−1)≤i
From the induction hypothesis on l − 1, the code C  is (l − 1)-extendable by adding the column h to G repeatedly. Hence, C is l-extendable and an l-extension of C can be obtained by adding the defining vector of H to G repeatedly l times. 2 Lemma 2.4. Let C be an [n, k, d]q code with generator matrix G, k ≥ 3. If F contains l distinct hyperplanes H1 , . . . , Hl of Σ through a (k − 3)-flat Δ with Δ ⊂ FC (i), d+l≤i
where 1 ≤ l < q, then C is l-extendable and an l-extension of C can be obtained by adding the defining vectors of H1 , . . . , Hl to G. Proof. We proceed by induction on l. The lemma follows from Lemma 2.1 for l = 1. Assume our assertion holds for l − 1 with l ≥ 2. Let H1 , . . . , Hl be hyperplanes of Σ  contained in F such that Δ = Hj is a (k − 3)-flat contained in FC (i). Let 1≤j≤l

d+l≤i
C  be the extension of C with generator matrix G = [G, hT 1 ], where h1 is the defining vector of H1 . Since wG (P ) = wG (P ) for P ∈ H1 and wG (P ) = wG (P ) + 1 for P ∈ / H1 , we have Hj ⊂ Σ \ FC  (d + 1 + q) for 2 ≤ j ≤ l and Δ⊂



FC  (i).

d+1+(l−1)≤i
From the induction hypothesis on l −1, the code C  is (l −1)-extendable and its extension can be obtained by adding the defining vectors of H2 , . . . , Hl to G. This completes the proof. 2 Lemma 2.5. Let C be an [n, k, d]q code with generator matrix G, k ≥ 3. If F contains l distinct hyperplanes H1 , . . . , Hl of Σ, 3 ≤ l < q, through a (k − 4)-flat S such that (a) S ⊂

d+l≤i
FC (i),

(b) Hα ∩ Hβ ∩ Hγ = S for any α, β, γ with 1 ≤ α < β < γ ≤ l,

H. Kanda et al. / Finite Fields and Their Applications 35 (2015) 159–171

(c) Hα ∩ Hβ ⊂



165

FC (i) for any α, β with 1 ≤ α < β ≤ l,

d+2≤i
then C is l-extendable by adding the defining vectors of H1 , . . . , Hl to G. Proof. We proceed by induction on l. Assume l = 3. Then, C is extendable by Corollary 2.2. Let C1 be the extension of C with generator matrix G1 = [G, hT 1 ], where h1 is the defining vector of H1 . Since wG1 (P ) = wG (P ) for P ∈ H1 and wG1 (P ) = wG (P ) + 1 for P ∈ / H1 , it holds that Hj ⊂ Σ \ FC1 (d + 1 + q) for j = 2, 3. Moreover, the conditions (a) and (c) imply H2 ∩ H3 ⊂ FC1 (i). Hence, we can apply Lemma 2.4 to H2 (d+1)+2≤i
and H3 so that C1 is 2-extendable. So, C is 3-extendable by adding the defining vectors of H1 , H2 , H3 to G. Assume our assertion holds for l − 1 with l ≥ 4. Let C  be the extension of C with generator matrix G = [G, hT ], where h is the defining vector of H1 . Since wG (P ) = wG (P ) for P ∈ H1 and wG (P ) = wG (P ) + 1 for P ∈ / H1 , it holds that Hj ⊂ Σ \ FC  (d + 1 + q) for 2 ≤ j ≤ l and that S ⊂ FC  (i). Since l ≥ 4, d+1+l−1≤i
the condition (b) holds for 2 ≤ α < β < γ ≤ l. Furthermore, from the conditions (a)–(c), we have Hα ∩ Hβ ⊂ FC  (i) for 2 ≤ α < β ≤ l. Hence, from the induction d+1+2≤i
hypothesis on l −1, C  is (l −1)-extendable. Thus C is l-extendable by adding the defining vectors of H1 , H2 , . . . , Hl to G. 2 The following “line condition” determines all possible lines in Σ. Lemma 2.6. (See [14].) For a line L = {P0 , P1 , . . . , Pq } in Σ, wG (L) :=

q 

wG (Pi ) ≡ 0 (mod q).

(2.1)

i=0

From now on, we only consider the case when q = 4. We set Fe = FC (0) ∪ FC (2). If a line L meets Fe in exactly i points, L is called an i-line. By Lemma 2.6, there exist only 1-lines, 3-lines or 5-lines of Fe . Such a set in Σ is called an odd set or a set of odd type [3]. Lemma 2.7. The set Fe forms an odd set in Σ. Theorems 1.3 and 1.4 can be proved applying the following two lemmas. Lemma 2.8. (See [12].) The set FC (0) forms an odd set in Fe if Fe is a flat in Σ. Lemma 2.9. (See [13].) Fe contains a hyperplane of Σ for k ≥ 3 if |Fe | < θk−2 + 2 · 4k−3 or |Fe | > θk−2 + 2 · 4k−2 − 4.

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A set B in PG(r, q) is called a blocking set with respect to s-flats if every s-flat in PG(r, q) meets B in at least one point. Theorem 2.10. (See [1].) Let B be a blocking set with respect to s-flats in PG(r, q). Then, |B| ≥ θr−s , where the equality holds if and only if B is an (r − s)-flat. The set Fe is a blocking set with respect to lines from Lemma 2.7, and a hyperplane forms an odd set. Hence, we get the following two lemmas by Theorem 2.10. Lemma 2.11. Fe contains at least θk−2 points. Lemma 2.12. If |Fe | = θk−2 , then Fe is a hyperplane of Σ. A t-flat Π of Σ with |Π ∩ FC (0)| = h, |Π ∩ FC (1)| = i, |Π ∩ FC (2)| = j is called an (h, i, j)t flat. An (h, i, j)1 flat is called an (h, i, j)-line. An (h, i, j)-plane, an (h, i, j)-solid and so on are defined similarly. The following lemma, originally proved for odd d [11], holds even when d ≡ 2 (mod 4). Lemma 2.13. (See [11].) |δ ∩ Fe | ∈ {5, 9, 13, 21} for any plane δ in Σ. Lemma 2.14. Let C be an [n, k, d]4 code with diversity (Φ0 , Φ1 ), k ≥ 4. Then, Φ0 is odd. Proof. Let Δ be a solid in Σ. From Table 19.10 in [3], Δ contains at least six 5-lines of Fe since Fe is an odd set. Let  be an (l0 , l1 , l2 )-line with l0 + l2 = 5. Then, l0 ∈ {1, 3, 5} [11]. Let δm be a (φ0,m , φ1,m , φ2,m )-plane for 1 ≤ m ≤ θk−3 through . Then, φ0,m is odd for all m [11]. Hence, 

θk−3

Φ0 =

(φ0,m − l0 ) + l0 ≡ l0 ≡ 1

(mod 2).

m=1

So, Φ0 is odd. 2 Although the following lemma is proved in [11] for odd d, the proof is also valid when d ≡ 2 (mod 4). Lemma 2.15. (See [11].) Φ0 ≥ θk−4 . If d is odd, Theorem 1.10 follows from Theorems 1.1, 1.9, Lemmas 2.14 and 2.15. If d ≡ 2 (mod 4), we get the following two lemmas. Lemmas 2.16 and 2.17 follow from Lemmas 2.3 and 2.4, respectively. Lemma 2.16. Let C be an [n, k, d]4 code with generator matrix G, k ≥ 3, d ≡ 2 (mod 4). If FC (0) ∪ FC (1) contains a hyperplane H of Σ, then C is 2-extendable by adding the defining vector of H twice to G.

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Lemma 2.17. Let C be an [n, k, d]4 code with generator matrix G, k ≥ 3, d ≡ 2 (mod 4). If F contains two distinct hyperplanes H1 , H2 of Σ meeting in a (k − 3)-flat Δ with Δ ⊂ FC (0) ∪ FC (1), then C is 2-extendable by adding the defining vectors of H1 , H2 to G. If d ≡ 1 (mod 4), we get the following three lemmas. Lemmas 2.18, 2.19 and 2.20 follow from Lemmas 2.3, 2.4 and 2.5 respectively. Lemma 2.18. Let C be an [n, k, d]4 code with generator matrix G, k ≥ 3, d ≡ 1 (mod 4). If FC (0) contains a hyperplane H of Σ, then C is 3-extendable by adding the defining vector of H repeatedly three times to G. Lemma 2.19. Let C be an [n, k, d]4 code with generator matrix G, k ≥ 3, d ≡ 1 (mod 4). If Fe contains three distinct hyperplanes H1 , H2 , H3 of Σ through a (k − 3)-flat Δ with Δ ⊂ FC (0), then C is 3-extendable by adding the defining vectors of H1 , H2 , H3 to G. Lemma 2.20. Let C be an [n, k, d]4 code with generator matrix G, k ≥ 3, d ≡ 1 (mod 4). If F contains three distinct hyperplanes H1 , H2 , H3 of Σ meeting in a (k − 4)-flat S with S ⊂ FC (0) satisfying (H1 ∩ H2 ) ∪ (H2 ∩ H3 ) ∪ (H3 ∩ H1 ) ⊂ FC (0) ∪ FC (3), then C is 3-extendable by adding the defining vectors of H1 , H2 , H3 to G. For given flats A1 , A2 , . . . , As in Σ, A1 , A2 , . . . , As denotes the smallest flat containing A1 , A2 , . . . , As . Proof of Theorem 1.5. Let C be an [n, k, d]4 code with k ≥ 3, d ≡ 1 (mod 4). (1) Assume Φ0 = θk−4 . Then, it follows from Lemma 4.4 in [11] that Φ1,2 = |FC (2)| = 9 · 4k−3 , Φ1,3 = |FC (3)| = 3 · 4k−3 or 9 · 4k−3 and that there is a (k − 4)-flat S contained in FC (0), which is called axis in [11]. Note that S = FC (0) in this case, for Φ0 = θk−4 . Take a plane δ so that δ ∩ S = ∅. Since δ has no point of FC (0), δ is a (0, u, 9)-plane with u = 3 or 9, see Table 3.3 in [11]. When u = 3, it also follows from Table 3.3 in [11] that δ contains three non-concurrent (0, 2, 3)-lines any two of which meet in a point of FC (1). So, one can take three non-concurrent (0, 0, 3)-lines on δ any two of which meet in a point of FC (3). It can be also shown that such three (0, 0, 3)-lines exist when u = 9. Thus, δ contains three non-concurrent (0, 0, 3)-lines, say 1 , 2 and 3 , any two of which meet in a point of FC (3). Let R1 = 2 ∩ 3 , R2 = 3 ∩ 1 and R3 = 1 ∩ 2 . For any point P ∈ S, the plane P, i is a (1, 0, 12)-plane for 1 ≤ i ≤ 3 since it has only one point from FC (0), see Table 3.4 in [11]. Hence, from the spectrum of a (1, 0, 12)-plane, the lines through P on the plane P, 1 consist of two (1, 0, 0)-lines and three (1, 0, 4)-lines. Taking three hyperplanes Hr = S, r , r = 1, 2, 3, we have (H1 ∩ H2 ) ∪ (H2 ∩ H3 ) ∪ (H3 ∩ H1 ) =



S, Ri ⊂ FC (0) ∪ FC (3),

1≤i≤3

for R1 , R2 , R3 ∈ FC (3). Applying Lemma 2.20, C is 3-extendable.

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(2) Assume Φ1,2 = 0. Then there is a (θk−2 , 0, 0)-hyperplane from Lemmas 3.1 and 3.2 in [8]. Hence, C is 3-extendable by Lemma 2.18. (3) Assume Φ1,3 = 0. Then it can be proved from Lemmas 4.1–4.3 in [8] that either FC (0) forms a hyperplane or Fe consists of three distinct hyperplanes through a fixed (k − 3)-flat contained in FC (0). Hence, C is 3-extendable by Lemmas 2.18 and 2.19. 2 Example 2.21. We investigate the code C1 in Example 1.11 from our geometrical point of view. We use the notations S, Ri , r , Hr and δ as in the proof of Theorem 1.5. From the generator matrix G1 , we can find the line

S = {(3, 2, 1, 0, 0), (0, 2, 1, 2, 1), (1, 1, 3, 2, 1), (2, 3, 2, 2, 1), (3, 0, 0, 2, 1)}. We can take R1 = (0, 0, 2, 1, 0), R2 = (0, 0, 3, 1, 0), R3 = (3, 2, 2, 1, 1) so that 1 , 2 and 3 are (0, 0, 3)-lines. Then, as the defining vectors of H1 , H2 , H3 , we obtain (2, 3, 0, 0, 1), (3, 0, 2, 1, 0), (3, 2, 1, 2, 1) which are to be added to G1 to get a 3-extension of C1 . Proof of Theorem 1.6. Let C be an [n, k, d]4 code with k ≥ 3, d ≡ 2 (mod 4), Φ1 = 4k−1 , and Φ0 < θk−3 + 2 · 4k−4 or Φ0 > θk−3 + 2 · 4k−3 − 4. Since |Fe | = θk−1 − Φ1 = θk−2 , Fe is a hyperplane of Σ by Lemma 2.12. Hence, by Lemma 2.8, FC (0) forms an odd set in Fe . Since Φ0 < θk−3 + 2 · 4k−4 or Φ0 > θk−3 + 2 · 4k−3 − 4, FC (0) contains a (k − 3)-flat, say π, by Lemma 2.9. Let H1 and H2 be distinct hyperplanes through π other than Fe . Then H1 and H2 contain no points of FC (2) and satisfy H1 ∩ H2 = π. Hence, C is 2-extendable by Lemma 2.17. 2 Proof of Theorem 1.7. Let C be an [n, k, d]4 code with k ≥ 3, d ≡ 2 (mod 4), Φ0 = θk−2 + 4k−2 and Φ1 = 2 · 4k−2 . Assume Φ1,3 = 0. Then there is a (θk−2 , 0, 0)-hyperplane from Remark (2) in [8]. Hence our assertion follows from Lemma 2.16. 2 Now, let C be an [n, k, d]4 code with k ≥ 3, d ≡ 2 (mod 4). A line  with | ∩FC (0)| = i, | ∩ F1 | = j is called an (i, j)-line. Let Λ1 be the set of all possible (i, j) for which an (i, j)-line exists in Σ. In the case Φ1,1 = 0 and Φ1,3 > 0, for any (i, j)-line  in Σ, we have 

wG (P ) = 0i + 3j + 2(θ1 − (i + j)) ≡ 0 (mod 4)

P ∈

by Lemma 2.6. Hence, 2i − j ≡ 2 (mod 4). This yields Lemma 2.22.

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Lemma 2.22. (See [14].) If Φ1,1 = 0 and Φ1,3 > 0 then, Λ1 = {(1, 0), (0, 2), (2, 2), (3, 0), (1, 4), (5, 0)}. Lemma 2.23. (See [14].) If all possible lines in Σ are given as in Lemma 2.22, then (1) there is a (θk−3 , 4k−2 )-hyperplane through a (k − 3)-flat π with π ⊂ FC (0); (2) for Q ∈ F1 , Q, π is a (θk−3 , 4k−2 )-hyperplane. Proof of Theorem 1.8. Assume Φ1,1 > 0 and Φ1,3 = 0. By Lemma 2.6, for any (i, j)-line  in Σ, we have 

wG (P ) = 0i + j + 2(θ1 − (i + j)) ≡ 0

(mod 4).

P ∈

Hence, 2i + j ≡ 2 (mod 4), which implies 2i − j ≡ 2 (mod 4). Hence, Λ1 is just the same with Λ1 in Lemma 2.22. From Lemma 2.23, F = FC (0) ∪ FC (1) contains a (θk−3 , 4k−2 )-hyperplane H1 through a (k − 3)-flat π with π ⊂ FC (0). Since H1 contains some (1, 4)-lines, take a (1, 4)-line 1 on H1 and let δ be a plane through 1 not contained in H1 . From Table 1 in [14], there are at least two (1, 4)-lines on δ. So, we can take a point Q ∈ F1 not contained in H1 . Then, we get another (θk−3 , 4k−2 )-hyperplane H2 = Q , π through π by part (2) of Lemma 2.23. Hence, C is 2-extendable by Lemma 2.17. We can also prove the 2-extendability for the case Φ1,1 = 0 and Φ1,3 > 0, similarly. 2 3. Proof of Theorem 1.9 In this section, let C be an [n, k, d]4 code with generator matrix G, d ≡ 0 (mod 4), k ≥ 4 and θk−4 < Φ0 ≤ θk−3 . First, we consider the case without 3-lines in the odd set Fe . Lemma 3.1. (See [5].) Let S be a proper subset of PG(r, q). Then S is a hyperplane of PG(r, q) if and only if every line in PG(r, q) meets S in one point or in θ1 = q + 1 points. Lemma 3.2. If there is no 3-line of Fe in Σ, then Fe is a hyperplane of Σ. Proof. Assume there is no 3-line of Fe . Then, possible lines are 1-lines or 5-lines since Fe forms an odd set in Σ. Hence, Fe is a hyperplane of Σ or Fe = Σ by Lemma 3.1. Suppose Fe = Σ. Then FC (0) forms an odd set in Σ by Lemma 2.8. Hence, FC (0) contains at least θk−2 points by Theorem 2.10, which contradicts to θk−4 < Φ0 ≤ θk−3 . 2 Recall that FC (0) forms an odd set in Fe when Fe is a flat of Σ by Lemma 2.8. It follows from θk−4 < Φ0 ≤ θk−3 that FC (0) is a (k−3)-flat and Φ0 = θk−3 by Lemma 2.11.

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When d is odd, Fe is a hyperplane of Σ such that Fe ∩FC (d) = ∅. Hence, C is extendable by Corollary 2.2. When d ≡ 2 (mod 4), let H1 and H2 be two hyperplanes of Σ through FC (0) other than Fe . Since Fe is a hyperplane of Σ, H1 and H2 contain no point from FC (2) = Fe \ FC (0)(⊃ FC (d)). Hence, C is 2-extendable by Lemma 2.17. Now we get the following two lemmas. Lemma 3.3. If there is no 3-line of Fe in Σ, then C is extendable. Lemma 3.4. If there is no 3-line of Fe in Σ, then C is 2-extendable when d is even. Next, we consider the case with a 3-line in Fe . If there is a (1, 1, 2)-line, any plane through the 3-line contains at least two points of FC (0) from Table 3.3 [11]. So, Φ0 ≥ (2 − 1) × θk−3 + 1 = θk−3 + 1, which contradicts to θk−4 < Φ0 ≤ θk−3 . Thus there is no (1, 1, 2)-line. The nonexistence of a (1, 1, 2)-line implies the nonexistence of an (h, i, j)-plane for (h, i, j) ∈ {(2, 3, 7), (2, 5, 7), (2, 7, 7), (2, 9, 7), (4, 3, 5), (4, 5, 5), (4, 7, 5), (4, 9, 5), (6, 3, 3), (6, 9, 3)} from Table 3.3 in [11]. If there is a (2, 2, 1)-line, any plane through the 3-line contains at least six points of FC (0) from Table 3.3 [11]. So, Φ0 ≥ (6 − 2) × θk−3 + 2 = 4 × θk−3 + 2, a contradiction. We can rule out a (2, 0, 1)-line and a (3, 1, 0)-line similarly. Thus we get the following three lemmas from Tables 2.1, 3.3 and 3.4 in [11]. Lemma 3.5. The possible 3-lines of Fe are (0, 0, 3)-lines and (0, 2, 3)-lines. Lemma 3.6. The possible 9-planes of Fe are (0, 3, 9)-planes and (0, 9, 9)-planes. Lemma 3.7. The possible 13-planes of Fe are (1, 0, 12)-planes and (1, 8, 12)-planes. From Lemmas 3.6 and 3.7, we get the following. Lemma 3.8. Any plane through a 3-line of Fe contains at most one point of FC (0). Let  be a 3-line of Fe . Then  is either a (0, 0, 3)-line or a (0, 2, 3)-line. Take two points P1 , P2 ∈ FC (0) and let m = P1 , P2 be the line through P1 , P2 , it follows from Lemma 3.8 that  and m are mutually disjoint. Hence, Δ = , m is a solid. Let δ = Q, m for Q ∈  ∩ FC (2), then δ ⊂ Δ. Since δ contains at least two points of FC (0), δ is not a 9-plane nor a 13-plane of Fe by Lemmas 3.6 and 3.7. Since δ contains two (1, 0, 4)-lines P1 , Q and P2 , Q , δ is not a 5-plane of Fe . Hence, δ is a 21-plane of Fe . Counting the number of points of FC (0) on the planes through  in Δ, we have |Δ ∩ FC (0)| ≤ 5. Since δ is a 21-plane, |δ ∩ FC (0)| ≥ 5 from Table 3.1 [11]. Hence, |Δ ∩ FC (0)| = |δ ∩ FC (0)| = 5. Since a (5, 0, 16)-plane contains a (5, 0, 0)-line from Table 3.1 [11], m is a (5, 0, 0)-line. This implies the following. Lemma 3.9. For any P1 , P2 ∈ FC (0), the line P1 , P2 is a (5, 0, 0)-line.

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It follows from Lemma 3.9 that FC (0) is a (k − 3)-flat, say π, since θk−4 < Φ0 ≤ θk−3 . Note that  contains three points of FC (2) and two points of FC (1) or FC (3). Let HQ = Q, π for Q ∈ FC (2). Then HQ is a (θk−3 , 0, 4k−2 )-hyperplane since the possible lines through Q ∈ FC (2) in HQ are (1, 0, 4)-lines from Lemmas 3.5 and 3.9. Since the number of planes through  is equal to |Φ0 |, all the planes through  must be 13-planes in Lemma 3.7. Counting the number of points of FC (2) on the planes through  by Lemma 3.7, we have |FC (2)| = (12 − 3) × θk−3 + 3 = 3 · 4k−2 . Hence, there are exactly three (θk−3 , 0, 4k−2 )-hyperplanes through π. The other hyperplanes through π can be obtained as HR = R, π with R ∈ FC (1) ∪ FC (3) and HR = R , π with R ∈ (FC (1) ∪ FC (3)) \ HR satisfying HR ∩ FC (2) = ∅ and HR ∩ FC (2) = ∅. Hence, C is extendable. Now we get the following. Lemma 3.10. If there is a 3-line of Fe in Σ, then C is extendable. Applying Lemma 2.17, we also get the following. Lemma 3.11. If there is a 3-line of Fe in Σ, then C is 2-extendable when d is even. When d ≡ 1 (mod 4), let H1 , H2 , H3 be the three (θk−3 , 0, 4k−2 )-hyperplanes through π. Applying Lemma 2.19, we get the following. Lemma 3.12. If there is a 3-line of Fe in Σ, then C is 3-extendable when d ≡ 1 (mod 4). Now, Theorem 1.9 follows from Lemmas 3.3, 3.4, 3.10, 3.11 and 3.12. 2 References [1] R.C. Bose, R.C. Burton, A characterization of flat spaces in a finite projective geometry and the uniqueness of the Hamming and the MacDonald codes, J. Comb. Theory 1 (1966) 96–104. [2] R. Hill, An extension theorem for linear codes, Des. Codes Cryptogr. 17 (1999) 151–157. [3] J.W.P. Hirschfeld, Finite Projective Spaces of Three Dimensions, Clarendon Press, Oxford, 1985. [4] J.W.P. Hirschfeld, Projective Geometries over Finite Fields, 2nd ed., Clarendon Press, Oxford, 1998. [5] T. Maruta, On the extendability of linear codes, Finite Fields Appl. 7 (2001) 350–354. [6] T. Maruta, Extendability of linear codes over GF(q) with minimum distance d, gcd(d, q) = 1, Discrete Math. 266 (2003) 377–385. [7] T. Maruta, Extendability of ternary linear codes, Des. Codes Cryptogr. 35 (2005) 175–190. [8] T. Maruta, Extendability of quaternary linear codes, Discrete Math. 293 (2005) 195–203. [9] T. Maruta, Extendability of linear codes over Fq , in: Proc. 11th International Workshop on Algebraic and Combinatorial Coding Theory, ACCT, Pamporovo, Bulgaria, 2008, pp. 203–209. [10] T. Maruta, Extension theorems for linear codes over finite fields, J. Geom. 101 (2011) 173–183. [11] T. Maruta, M. Takeda, K. Kawakami, New sufficient conditions for the extendability of quaternary linear codes, Finite Fields Appl. 14 (2008) 615–634. [12] T. Tanaka, T. Maruta, Classification of the odd sets in PG(4, 4) and its application to coding theory, Appl. Algebra Eng. Commun. Comput. 24 (2013) 176–196. [13] T. Tanaka, T. Maruta, A characterization of some odd sets in projective space of order 4 and the extendability of quaternary linear codes, J. Geom. 105 (2014) 79–86. [14] Y. Yoshida, T. Maruta, An extension theorem for [n, k, d]q codes with gcd(d, q) = 2, Australas. J. Comb. 48 (2010) 117–131.