# Cyclotomic Trace Codes

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

- $\left|\mathcal{P}\right|=v$,
- every element of $\mathcal{B}$ is incident with exactly k elements of $\mathcal{P}$,
- every t elements of $\mathcal{P}$ are incident with exactly $\lambda $ elements of $\mathcal{B}$.

## 3. Construction of Linear Codes

#### 3.1. Codes Obtained from Fields of Odd Order

**Result**

**1**

**.**Let D be the set of all quadratic residues in $GF{\left({p}^{m}\right)}^{*}$, where p is an odd prime. If m is odd, then ${C}_{D}$ is a one-weight code over $GF\left(p\right)$ with parameters $\left[\right(q-1)/2,m,(p-1)q/2p]$. If m is even, then ${C}_{D}$ is a two-weight code over $GF\left(p\right)$ with parameters $[(q-1)/2,m,(p-1)(q-\sqrt{q})/2p]$ and weight enumerator $1+\frac{q-1}{2}{z}^{(p-1)(q-\sqrt{q})/2p}+\frac{q-1}{2}{z}^{(p-1)(q+\sqrt{q})/2p}$.

**Remark**

**2.**

**Remark**

**3.**

**Remark**

**4.**

**Remark**

**5.**

#### 3.2. Codes Obtained from Fields of Even Order

**Remark**

**6.**

**Remark**

**7.**

## 4. Support Designs and Generalized Block Graphs

**Remark**

**9.**

**Remark**

**10.**

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Ding, C. A construction of binary linear codes from Boolean functions. Discret. Math.
**2016**, 339, 2288–2303. [Google Scholar] [CrossRef] [Green Version] - Ding, C.; Niederreiter, H. Cyclotomic linear codes of order 3. IEEE Trans. Inf. Theory
**2007**, 53, 2274–2277. [Google Scholar] [CrossRef] - Ding, C.; Luo, J.; Niederreiter, H. Two weight codes punctured from irreducible cyclic codes. In Proceedings of the First International Workshop on Coding Theory and Cryptography, Wuyi Mountain, China, 11–15 June 2007; pp. 119–124. [Google Scholar]
- Shi, M.; Qian, L.; Solé, P. Few-weight codes from trace codes over a local ring. Appl. Algebra Eng. Commun. Comput.
**2018**, 29, 335–350. [Google Scholar] [CrossRef] - Wang, X.; Zheng, D.; Liu, H. Several classes of linear codes and their weight distributions. Appl. Algebra Eng. Commun. Comput.
**2019**, 30, 75–92. [Google Scholar] [CrossRef] - Zhou, Z. Three-weight ternary linear codes from a family of cyclic difference sets. Des. Codes Cryptogr.
**2018**, 86, 2513–2523. [Google Scholar] [CrossRef] - Ding, C. Linear Codes from Some 2-Designs. IEEE Trans. Inf. Theory
**2015**, 61, 3265–3275. [Google Scholar] [CrossRef] - Olmez, O. A link between combinatorial designs and three-weight linear codes. Des. Codes Cryptogr.
**2018**, 86, 817–833. [Google Scholar] [CrossRef] - Bonisoli, A. Every equidistant code is a sequence of dual Hamming codes. Ars Combinatoria
**1983**, 18, 181–186. [Google Scholar] - Jungnickel, D.; Tonchev, V.D. The classification of antipodal two-weight linear codes. Finite Fields Appl.
**2018**, 50, 372–381. [Google Scholar] [CrossRef] - Jungnickel, D.; Tonchev, V.D. On Bonisoli’s theorem and the block codes of Steiner triple systems. Des. Codes Cryptogr.
**2018**, 86, 449–462. [Google Scholar] [CrossRef] - Ding, C.; Munemasa, A.; Tonchev, V.D. Bent Vectorial Functions, Codes and Designs. IEEE Trans. Inf. Theory
**2019**. [Google Scholar] [CrossRef] - Brouwer, A.E. Some new two-weight codes and strongly regular graphs. Discrete Appl. Math.
**1985**, 10, 111–114. [Google Scholar] [CrossRef] [Green Version] - Crnković, D.; Mikulić, V. Self-orthogonal doubly-even codes from Hadamard matrices of order 48. Adv. Appl. Discret. Math.
**2008**, 1, 159–170. [Google Scholar] - Bosma, W.; Cannon, J. Handbook of Magma Functions; Department of Mathematics, University of Sydney: Sydney, NWS, Australia, 1994. [Google Scholar]
- Huffman, W.C.; Pless, V. Fundamentals of Error-Correcting Codes; Cambridge University Press: Cambridge, UK, 2003. [Google Scholar]
- Grassl, M. Bounds on the Minimum Distance of Linear Codes and Quantum Codes. Available online: http://www.codetables.de (accessed on 11 August 2019).
- Brouwer, A.E. Strongly Regular Graphs. In Handbook of Combinatorial Designs, 2nd ed.; Colbourn, C.J., Dinitz, J.H., Eds.; Chapman & Hall/CRC: Boca Raton, FL, USA, 2007; pp. 852–868. [Google Scholar]
- Shrikhande, M.S.; Sane, S.S. Quasi-Symmetric Designs; Cambridge University Press: Cambridge, UK, 1991. [Google Scholar]
- Tonchev, V.D. Combinatorial Configurations: Designs, Codes, Graphs; John Willey & Sons: New York, NY, USA, 1988. [Google Scholar]
- Brouwer, A.E.; Cohen, A.M.; Neumaier, A. Distance-Regular Graphs; Springer: Berlin, Germany, 1989. [Google Scholar]
- Brouwer, A.E. Parameters of Strongly Regular Graphs. Available online: http://www.win.tue.nl/$\sim$aeb/graphs/srg/srgtab.html (accessed on 24 May 2019).
- Calderbank, A.R.; Kantor, W.M. The geometry of two-weight codes. Bull. Lond. Math. Soc.
**1986**, 118, 97–122. [Google Scholar] [CrossRef] - Brouwer, A.E.; Haemers, W.H. Structure and uniqueness of the (81,20,1,6) strongly regular graph. Discret. Math.
**1992**, 106/107, 77–82. [Google Scholar] [CrossRef] - Calderbank, A.R.; Rains, E.M.; Shor, P.W.; Sloane, N.J.A. Quantum error correction via codes over GF(4). IEEE Trans. Inf. Theory
**1998**, 44, 1369–1387. [Google Scholar] [CrossRef] - Shor, P.W. Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A
**1995**, 52, R2493. [Google Scholar] [CrossRef] [PubMed] - Tonchev, V.D. Quantum codes from caps. Discret. Math.
**2008**, 308, 6368–6372. [Google Scholar] [CrossRef] [Green Version] - Ionin, Y.J.; van Trung, T. Symmetric Designs. In Handbook of Combinatorial Designs, 2nd ed.; Colbourn, C.J., Dinitz, J.H., Eds.; Chapman & Hall/CRC: Boca Raton, FL, USA, 2007; pp. 110–124. [Google Scholar]
- Shrikhande, M.S. Quasi-Symmetric Designs. In Handbook of Combinatorial Designs, 2nd ed.; Colbourn, C.J., Dinitz, J.H., Eds.; Chapman & Hall/CRC: Boca Raton, FL, USA, 2007; pp. 578–582. [Google Scholar]

$\mathit{p},\mathit{q},\mathit{m}$ | Code | Weight Dist. | Corr. Proj. Code | SRG |
---|---|---|---|---|

3, 9, 2 | ${[4,2,2]}_{3}$ | $<0,1>,<2,4>,<4,4>$ | ${[2,2,1]}^{*}$, ${d}^{\perp}=2$ | (9,4,1,2) |

3, 81, 4 | ${[40,4,24]}_{3}$ | $<0,1>,<24,40>,<30,40>$ | ${[20,4,12]}^{*}$, ${d}^{\perp}=3$ | (81,40,19,20) |

3, 729, 6 | ${[364,6,234]}_{3}$ | $<0,1>,<234,364>,<252,364>$ | $[182,6,117]$, ${d}^{\perp}=3$ | (729,364,181,182) |

5, 25, 2 | ${[12,2,8]}_{5}$ | $<0,1>,<8,12>,<12,12>$ | ${[3,2,2]}^{*}$, ${d}^{\perp}=3$ | (25,12,5,6) |

5, 625, 4 | ${[312,4,240]}_{5}$ | $<0,1>,<240,312>,<260,312>$ | $[78,4,60]$, ${d}^{\perp}=3$ | (625,312,155,156) |

7, 49, 2 | ${[24,2,18]}_{7}$ | $<0,1>,<18,24>,<24,24>$ | ${[4,2,3]}^{*}$, ${d}^{\perp}=3$ | (49,24,11,12) |

7, 2401, 4 | ${[1200,4,1008]}_{7}$ | $<0,1>,<1008,1200>,<1050,1200>$ | $[200,4,168]$, ${d}^{\perp}=3$ | (2401,1200,599,600) |

$\mathit{p},\mathit{q},\mathit{m}$ | Code | Weight Dist. | Corr. Proj. Code | SRG |
---|---|---|---|---|

5, 25, 2 | ${[8,2,4]}_{5}$ | $<0,1>,<4,8>,<8,16>$ | ${[2,2,1]}^{*}$, ${d}^{\perp}=2$ | (25,8,3,2) |

5, 625, 4 | ${[208,4,160]}_{5}$ | $<0,1>,<160,416>,<180,208>$ | ${[52,4,40]}^{*}$, ${d}^{\perp}=3$ | (625,208,63,72) |

5, 15625, 6 | ${[5208,6,4100]}_{5}$ | $<0,1>,<4100,5208>,<4200,10416>$ | $[1302,6,1025]$, ${d}^{\perp}=3$ | (15625,5208,1763,1722) |

11, 121, 2 | ${[40,2,30]}_{11}$ | $<0,1>,<30,40>,<40,80>$ | ${[4,2,3]}^{*}$, ${d}^{\perp}=3$ | (121,40,15,12) |

11, 14641, 4 | ${[4880,4,4400]}_{11}$ | $<0,1>,<4400,9760>,<4510,4880>$ | $[488,4,440]$, ${d}^{\perp}=3$ | (14641,4880,1599,1640) |

17, 289, 2 | ${[96,2,80]}_{17}$ | $<0,1>,<80,96>,<96,192>$ | ${[6,2,5]}^{*}$, ${d}^{\perp}=3$ | (289,96,35,30) |

$\mathit{p},\mathit{q},\mathit{m}$ | Code | Weight Dist. | Corr. Proj. Code | SRG |
---|---|---|---|---|

3, 81, 4 | ${[20,4,12]}_{3}^{*}$ | $<0,1>,<12,60>,<18,20>$ | ${[10,4,6]}^{*}$, ${d}^{\perp}=4$ | (81,20,1,6) |

3, 729, 6 | ${[182,6,108]}_{3}$ | $<0,1>,<108,182>,<126,546>$ | $[91,6,54]$, ${d}^{\perp}=3$ | (729,182,55,42) |

5, 25, 2 | ${[6,2,4]}_{5},{d}^{\perp}=3$ | $<0,1>,<4,12>,<6,12>$ | (25,12,5,6) | |

5, 15625, 6 | ${[3906,6,3100]}_{5}^{*}$ | $<0,1>,<3100,7812>,<3150,7812>$ | $[1953,6,1550]$, ${d}^{\perp}=3$ | (15625,7812,3905,3906) |

7, 49, 2 | ${[12,2,6]}_{7}$ | $<0,1>,<6,12>,<12,36>$ | ${[2,2,1]}^{*}$, ${d}^{\perp}=2$ | (49,12,5,2) |

7, 2401, 4 | ${[600,4,504]}_{7}$ | $<0,1>,<504,1800>,<546,600>$ | ${[100,4,84]}^{*}$, ${d}^{\perp}=3$ | (2401,600,131,156) |

11, 121, 2 | ${[30,2,20]}_{11}$ | $<0,1>,<20,30>,<30,90>$ | ${[3,2,2]}^{*}$, ${d}^{\perp}=3$ | (121,30,11,6) |

11, 14641, 4 | ${[3660,4,3300]}_{11}$ | $<0,1>,<3300,10980>,<3410,3660>$ | $[366,4,330]$, ${d}^{\perp}=3$ | (14641,3660,869,871) |

13, 169, 2 | ${[42,2,36]}_{13}$ | $<0,1>,<36,84>,<42,84>$ | ${[7,2,6]}^{*}$, ${d}^{\perp}=3$ | (169,84,41,42) |

17, 289, 2 | ${[72,2,64]}_{17}$ | $<0,1>,<64,144>,<72,144>$ | ${[9,2,8]}^{*}$, ${d}^{\perp}=3$ | (289,144,71,72) |

19, 361, 2 | ${[90,2,72]}_{19}$ | $<0,1>,<72,90>,<90,270>$ | ${[5,2,4]}^{*}$, ${d}^{\perp}=3$ | (361,90,29,20) |

$\mathit{p},\mathit{q},\mathit{m}$ | Code | Weight Dist. | Corr. Proj. Code | SRG |
---|---|---|---|---|

3, 81, 4 | ${[16,4,6]}_{3}$ | $<0,1>,<6,16>,<12,64>$ | $[8,4,3]$, ${d}^{\perp}=3$ | (81,16,7,2) |

7, 2401, 4 | ${[480,4,378]}_{7}$ | $<0,1>,<378,480>,<420,1920>$ | $[80,4,63]$, ${d}^{\perp}=3$ | (2401,480,119,90) |

13, 28561, 4 | ${[5712,4,5148]}_{13}$ | $<0,1>,<5148,5712>,<5304,22848>$ | $[476,4,429]$, ${d}^{\perp}=3$ | (28561,5712,1223,1122) |

19, 361, 2 | ${[72,2,54]}_{19}$ | $<0,1>,<54,72>,<72,288>$ | ${[4,2,3]}^{*}$, ${d}^{\perp}=3$ | (361,72,23,12) |

$\mathit{p},\mathit{q},\mathit{m}$ | Code | Weight Dist. | Corr. Proj. Code | SRG |
---|---|---|---|---|

3, 9, 2 | ${[4,2,2]}_{3}$ | $<0,1>,<2,4>,<4,4>$ | ${[2,2,1]}^{*}$, ${d}^{\perp}=2$ | (9,4,1,2) |

3, 81, 4 | ${[40,4,24]}_{3}$ | $<0,1>,<24,40>,<30,40>$ | ${[20,4,12]}^{*}$, ${d}^{\perp}=3$ | (81,40,19,20) |

3, 729, 6 | ${[364,6,236]}_{3}$ | $<0,1>,<234,364>,<252,364>$ | $[182,6,117]$, ${d}^{\perp}=3$ | (729,364,181,182) |

5, 625, 4 | ${[104,4,80]}_{5}$ | $<0,1>,<80,520>,<100,104>$ | ${[26,4,20]}^{*}$, ${d}^{\perp}=4$ | (625,104,3,20) |

5, 15625, 6 | ${[2604,6,2000]}_{5}$ | $<0,1>,<2000,2604>,<2100,13020>$ | $[651,6,500]$, ${d}^{\perp}=3$ | (15625,2604,503,420) |

7, 49, 2 | ${[8,2,6]}_{7}$ | $<0,1>,<6,24>,<8,24>$ | ${[4,2,3]}^{*}$, ${d}^{\perp}=3$ | (49,24,11,12) |

7, 2401, 4 | ${[400,4,336]}_{7}$ | $<0,1>,<336,1200>,<350,1200>$ | $[200,4,168]$, ${d}^{\perp}=3$ | (2401,1200,599,600) |

11, 121, 2 | ${[20,2,10]}_{11}$ | $<0,1>,<10,20>,<20,100>$ | ${[2,2,1]}^{*}$, ${d}^{\perp}=2$ | (121,20,9,2) |

11, 14641, 4 | ${[2440,4,2200]}_{11}$ | $<0,1>,<2200,12200>,<2310,2440>$ | ${[244,4,220]}^{*}$, ${d}^{\perp}=3$ | (14641,2440,420,341) |

13, 169, 2 | ${[28,2,24]}_{13}$ | $<0,1>,<24,84>,<28,84>$ | ${[7,2,6]}^{*}$, ${d}^{\perp}=3$ | (169,84,41,42) |

13, 28561, 4 | ${[4760,4,4368]}_{13}$ | $<0,1>,<4368,14280>,<4420,14280>$ | $[1190,4,1092]$, ${d}^{\perp}=3$ | (28561,14280,7139,7140) |

17, 289, 2 | ${[48,2,32]}_{17}$ | $<0,1>,<32,48>,<48,240>$ | ${[3,2,2]}^{*}$, ${d}^{\perp}=3$ | (289,48,17,6) |

19, 361, 2 | ${[60,2,54]}_{19}$ | $<0,1>,<54,180>,<60,180>$ | ${[10,2,9]}^{*}$, ${d}^{\perp}=3$ | (361,180,89,90) |

$\mathit{p},\mathit{q},\mathit{m}$ | s | Code | Weight Dist. | SRG |
---|---|---|---|---|

2,16, 4 | 3,6 | ${[5,4,2]}_{2}^{*},{d}^{\perp}=5$ | $<0,1>,<2,10>,<4,5>$ | (16,5,0,2) |

2,64, 6 | 3,6 | ${[21,6,8]}_{2},{d}^{\perp}=3$ | $<0,1>,<8,21>,<12,42>$ | (64,21,8,6) |

2,256, 8 | 3,6 | ${[85,8,40]}_{2}^{*},{d}^{\perp}=3$ | $<0,1>,<40,170>,<48,85>$ | (256,85,24,30) |

2,256, 8 | 5 | ${[51,8,24]}_{2}^{*},{d}^{\perp}=3$ | $<0,1>,<24,204>,<32,51>$ | (256,51,2,12) |

2,1024, 10 | 3,6 | ${[341,10,160]}_{2},{d}^{\perp}=3$ | $<0,1>,<160,341>,<176,682>$ | (1024,341,120,110) |

2,4096, 12 | 3,6 | ${[1365,12,672]}_{2},{d}^{\perp}=3$ | $<0,1>,<672,2730>,<704,1365>$ | (4096,1365,440,462) |

2,4096, 12 | 5 | ${[819,12,348]}_{2},{d}^{\perp}=3$ | $<0,1>,<384,819>,<416,3276>$ | (4096,819,194,156) |

$\mathit{p},\mathit{q},\mathit{m}$ | s | Code | Weight Dist. | Corr. Proj. Code | SRG |
---|---|---|---|---|---|

4,64, 3 | 3 | ${[21,3,12]}_{4}$ | $<0,1>,<12,21>,<18,42>$ | ${[7,3,4]}^{*}$, ${d}^{\perp}=3$ | (64,21,8,6) |

4,64, 3 | 9 | ${[7,3,4]}_{4}^{*}$ | $<0,1>,<4,21>,<6,42>$, ${d}^{\perp}=3$ | (64,21,8,6) | |

4,256, 4 | 5 | ${[51,4,36]}_{4}$ | $<0,1>,<36,204>,<48,51>$ | ${[17,4,12]}^{*}$, ${d}^{\perp}=4$ | (256,51,2,12) |

4,256, 4 | 15 | ${[17,4,12]}_{4}^{*}$ | $<0,1>,<12,204>,<6,51>$, ${d}^{\perp}=4$ | (256,51,2,12) | |

4,1024, 5 | 11 | ${[93,5,48]}_{4}$ | $<0,1>,<48,93>,<72,930>$ | $[31,5,16]$, ${d}^{\perp}=3$ | (1024,93,32,6) |

4,1024, 5 | 33 | ${[31,5,16]}_{4}$ | $<0,1>,<16,93>,<24,930>$, ${d}^{\perp}=3$ | (1024,93,32,6) | |

4,4096, 6 | 3 | ${[1365,6,1008]}_{4}$ | $<0,1>,<1008,2730>,<1056,1365>$ | $[455,6,336]$, ${d}^{\perp}=3$ | (4096,1365,440,462) |

4,4096, 6 | 5 | ${[819,6,576]}_{4}$ | $<0,1>,<576,819>,<624,3276>$ | $[273,6,192]$, ${d}^{\perp}=3$ | (4096,819,194,156) |

4,4096, 6 | 9 | ${[455,6,336]}_{4}$ | $<0,1>,<1336,2730>,<352,1365>$, ${d}^{\perp}=3$ | (4096,1365,440,462) | |

4,4096, 6 | 13 | ${[315,6,192]}_{4}$ | $<0,1>,<192,315>,<240,3780>$ | $[105,6,64]$, ${d}^{\perp}=3$ | (4096,315,74,20) |

4,16384, 7 | 43 | ${[381,7,192]}_{4}$ | $<0,1>,<192,381>,<288,16002>$ | $[127,7,64]$, ${d}^{\perp}=3$ | (16384,381,128,6) |

4,16384, 7 | 129 | ${[127,7,64]}_{4}$ | $<0,1>,<64,381>,<96,16002>$, ${d}^{\perp}=3$ | (16384,381,128,6) |

$\mathit{p},\mathit{q},\mathit{m}$ | s | Code | Weight Dist. | Corr. Proj. Code | SRG |
---|---|---|---|---|---|

8,4096, 4 | 3 | ${[1365,4,1176]}_{8}$ | $<0,1>,<1176,2730>,<1232,1365>$ | $[195,4,168]$, ${d}^{\perp}=3$ | (4096,1365,440,462) |

8,4096, 4 | 5 | ${[819,4,672]}_{8}$ | $<0,1>,<672,819>,<728,3276>$ | $[117,4,96]$, ${d}^{\perp}=3$ | (4096,819,194,156) |

8,4096, 4 | 9 | ${[455,4,392]}_{8}$ | $<0,1>,<392,3640>,<448,455>$ | ${[65,4,56]}^{*}$, ${d}^{\perp}=4$ | (4096,455,6,56) |

8,4096, 4 | 13 | ${[315,4,224]}_{8}$ | $<0,1>,<224,315>,<280,3780>$ | $[45,4,32]$, ${d}^{\perp}=3$ | (4096,315,74,20) |

$\mathit{p},\mathit{q},\mathit{m}$ | s | Code | Weight Dist. | Support Design |
---|---|---|---|---|

3,81,4 | 4 | ${[10,4,6]}_{3}$ | $<0,1>,<6,60>,<9,20>$ | 3-(10,4,1), $b=30$ |

5,625,4 | 6 | ${[26,4,20]}_{5}$ | $<0,1>,<20,520>,<25,100>$ | 3-(26,6,1), $b=130$ |

4,64,3 | 9 | ${[7,3,4]}_{4}$ | $<0,1>,<4,21>,<6,42>$ | 2-(7,3,1), $b=7$ |

4,256,4 | 5 | ${[17,4,12]}_{4}$ | $<0,1>,<12,204>,<16,51>$ | 3-(17,5,1), $b=68$ |

4,1024,5 | 11 | ${[31,5,16]}_{4}$ | $<0,1>,<16,93>,<24,930>$ | 2-(31,15,7), $b=31$ |

4,1024,5 | 11 | ${[31,5,16]}_{4}$ | $<0,1>,<16,93>,<24,930>$ | 2-(31,7,7), $b=155$ |

4,1024,5 | 31 | ${[11,5,6]}_{4}$ | $<0,1>,<6,155>,<7,165>,<8,165>,$ | 2-(11,5,10), $b=55$ |

$<9,330>,<10,165>,<11,33>$ | ||||

4,1024,5 | 31 | ${[11,5,6]}_{4}$ | $<0,1>,<6,155>,<7,165>,<8,165>,$ | 2-(11,3,3), $b=55$ |

$<9,330>,<10,165>,<11,33>$ | ||||

4,1024,5 | 31 | ${[11,5,6]}_{4}$ | $<0,1>,<6,155>,<7,165>,<8,165>,$ | 2-(11,4,6), $b=55$ |

$<9,330>,<10,165>,<11,33>$ | ||||

4,16384,7 | 129 | ${[127,7,64]}_{4}$ | $<0,1>,<64,381>,<96,16002>$ | 2-(127,63,31), $b=127$ |

4,16384,7 | 129 | ${[127,7,64]}_{4}$ | $<0,1>,<16,93>,<24,930>$ | 2-(127,31,155), $b=2667$ |

8,4096,4 | 45 | ${[13,4,9]}_{8}$ | $<0,1>,<9,364>,<10,546>,$ | 2-(13,4,4), $b=52$ |

$<11,1092>,<12,1365>,<13,728>$ | ||||

8,4096,4 | 45 | ${[13,4,9]}_{8}$ | $<0,1>,<9,364>,<10,546>,$ | 2-(13,3,3), $b=78$ |

$<11,1092>,<12,1365>,<13,728>$ | ||||

8,4096,6 | 9 | ${[65,4,56]}_{8}$ | $<0,1>,<56,3640>,<64,455>$ | 3-(65,9,1), $b=520$ |

$\mathit{p},\mathit{q},\mathit{m}$ | s | Code | Weight Dist. | Gen. Block Graph |
---|---|---|---|---|

2,16,4 | 3,6 | ${[5,4,2]}_{2}$ | $<0,1>,<2,10>,<4,5>$ | SRG(10,3,0,1) |

3,81,4 | 4 | ${[10,4,6]}_{3}$ | $<0,1>,<6,60>,<9,20>$ | DRG, $v=30,[3,2,2,2;1,1,1,3]$ |

3,81,4 | 5 | ${[8,4,3]}_{3}$ | $<0,1>,<3,16>,<6,64>$ | SRG(16,6,2,2) |

4,256,4 | 5 | ${[17,4,12]}_{4}$ | $<0,1>,<12,204>,<16,51>$ | DRG, $v=68,[12,10,3;1,3,8]$ |

4,1024,5 | 11 | ${[31,5,16]}_{4}$ | $<0,1>,<16,93>,<24,930>$ | SRG(155,42,17,9) |

4,1024,5 | 31 | ${[11,5,6]}_{4}$ | $<0,1>,<6,155>,<7,165>,<8,165>,$ | SRG(55,18,9,4) |

$<9,330>,<10,165>,<11,33>$ | ||||

4,16384,7 | 129 | ${[127,7,64]}_{4}$ | $<0,1>,<64,381>,<96,16002>$ | SRG(2667,186,65,9) |

8,4096,4 | 45 | ${[13,4,9]}_{8}$ | $<0,1>,<9,364>,<10,546>,$ | SRG(78,22,11,4) |

$<11,1092>,<12,1365>,<13,728>$ | ||||

8,4096,4 | 15 | ${[39,4,30]}_{8}$ | $<0,1>,<30,546>,<33,1092>,$ | SRG(169,36,13,6) |

$<35,1092>,<36,1365>$ |

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**MDPI and ACS Style**

Crnković, D.; Švob, A.; Tonchev, V.D.
Cyclotomic Trace Codes. *Algorithms* **2019**, *12*, 168.
https://doi.org/10.3390/a12080168

**AMA Style**

Crnković D, Švob A, Tonchev VD.
Cyclotomic Trace Codes. *Algorithms*. 2019; 12(8):168.
https://doi.org/10.3390/a12080168

**Chicago/Turabian Style**

Crnković, Dean, Andrea Švob, and Vladimir D. Tonchev.
2019. "Cyclotomic Trace Codes" *Algorithms* 12, no. 8: 168.
https://doi.org/10.3390/a12080168