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## Non-existence of some 4-dimensional Griesmer codes over finite fields

#### Kazuki KUMEGAWA [1] , Tatsuya MARUTA [2]

We prove the non--existence of $[g_q(4,d),4,d]_q$ codes for $d=2q^3-rq^2-2q+1$ for $3 \le r \le (q+1)/2$, $q \ge 5$; $d=2q^3-3q^2-3q+1$ for $q \ge 9$; $d=2q^3-4q^2-3q+1$ for $q \ge 9$; and $d=q^3-q^2-rq-2$ with $r=4, 5$ or $6$ for $q \ge 9$, where $g_q(4,d)=\sum_{i=0}^{3} \left\lceil d/q^i \right\rceil$. This yields that $n_q(4,d) = g_q(4,d)+1$ for $2q^3-3q^2-3q+1 \le d \le 2q^3-3q^2$, $2q^3-5q^2-2q+1 \le d \le 2q^3-5q^2$ and $q^3-q^2-rq-2 \le d \le q^3-q^2-rq$ with $4 \le r \le 6$ for $q \ge 9$ and that $n_q(4,d) \ge g_q(4,d)+1$ for $2q^3-rq^2-2q+1 \le d \le 2q^3-rq^2-q$ for $3 \le r \le (q+1)/2$, $q \ge 5$ and $2q^3-4q^2-3q+1 \le d \le 2q^3-4q^2-2q$ for $q \ge 9$, where $n_q(4,d)$ denotes the minimum length $n$ for which an $[n,4,d]_q$ code exists.
Optimal linear codes, Griesmer bound, Arcs in PG$(rq)$
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Primary Language en Engineering Articles Author: Kazuki KUMEGAWA Orcid: 0000-0001-7858-0787Author: Tatsuya MARUTA (Primary Author) Publication Date : May 28, 2018
 Bibtex @research article { jacodesmath427968, journal = {Journal of Algebra Combinatorics Discrete Structures and Applications}, issn = {}, eissn = {2148-838X}, address = {}, publisher = {Yildiz Technical University}, year = {2018}, volume = {5}, pages = {101 - 116}, doi = {10.13069/jacodesmath.427968}, title = {Non-existence of some 4-dimensional Griesmer codes over finite fields}, key = {cite}, author = {Kumegawa, Kazuki and Maruta, Tatsuya} } APA Kumegawa, K , Maruta, T . (2018). Non-existence of some 4-dimensional Griesmer codes over finite fields . Journal of Algebra Combinatorics Discrete Structures and Applications , 5 (2) , 101-116 . DOI: 10.13069/jacodesmath.427968 MLA Kumegawa, K , Maruta, T . "Non-existence of some 4-dimensional Griesmer codes over finite fields" . Journal of Algebra Combinatorics Discrete Structures and Applications 5 (2018 ): 101-116 Chicago Kumegawa, K , Maruta, T . "Non-existence of some 4-dimensional Griesmer codes over finite fields". Journal of Algebra Combinatorics Discrete Structures and Applications 5 (2018 ): 101-116 RIS TY - JOUR T1 - Non-existence of some 4-dimensional Griesmer codes over finite fields AU - Kazuki Kumegawa , Tatsuya Maruta Y1 - 2018 PY - 2018 N1 - doi: 10.13069/jacodesmath.427968 DO - 10.13069/jacodesmath.427968 T2 - Journal of Algebra Combinatorics Discrete Structures and Applications JF - Journal JO - JOR SP - 101 EP - 116 VL - 5 IS - 2 SN - -2148-838X M3 - doi: 10.13069/jacodesmath.427968 UR - https://doi.org/10.13069/jacodesmath.427968 Y2 - 2018 ER - EndNote %0 Journal of Algebra Combinatorics Discrete Structures and Applications Non-existence of some 4-dimensional Griesmer codes over finite fields %A Kazuki Kumegawa , Tatsuya Maruta %T Non-existence of some 4-dimensional Griesmer codes over finite fields %D 2018 %J Journal of Algebra Combinatorics Discrete Structures and Applications %P -2148-838X %V 5 %N 2 %R doi: 10.13069/jacodesmath.427968 %U 10.13069/jacodesmath.427968 ISNAD Kumegawa, Kazuki , Maruta, Tatsuya . "Non-existence of some 4-dimensional Griesmer codes over finite fields". Journal of Algebra Combinatorics Discrete Structures and Applications 5 / 2 (May 2018): 101-116 . https://doi.org/10.13069/jacodesmath.427968 AMA Kumegawa K , Maruta T . Non-existence of some 4-dimensional Griesmer codes over finite fields. Journal of Algebra Combinatorics Discrete Structures and Applications. 2018; 5(2): 101-116. Vancouver Kumegawa K , Maruta T . Non-existence of some 4-dimensional Griesmer codes over finite fields. Journal of Algebra Combinatorics Discrete Structures and Applications. 2018; 5(2): 101-116.

Authors of the Article
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