Best Known (258−61, 258, s)-Nets in Base 4
(258−61, 258, 1040)-Net over F4 — Constructive and digital
Digital (197, 258, 1040)-net over F4, using
- 42 times duplication [i] based on digital (195, 256, 1040)-net over F4, using
- trace code for nets [i] based on digital (3, 64, 260)-net over F256, using
- net from sequence [i] based on digital (3, 259)-sequence over F256, using
- trace code for nets [i] based on digital (3, 64, 260)-net over F256, using
(258−61, 258, 3030)-Net over F4 — Digital
Digital (197, 258, 3030)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4258, 3030, F4, 61) (dual of [3030, 2772, 62]-code), using
- 2771 step Varšamov–Edel lengthening with (ri) = (17, 7, 4, 3, 2, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 9 times 0, 1, 9 times 0, 1, 9 times 0, 1, 10 times 0, 1, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 11 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 12 times 0, 1, 13 times 0, 1, 13 times 0, 1, 13 times 0, 1, 14 times 0, 1, 14 times 0, 1, 14 times 0, 1, 15 times 0, 1, 15 times 0, 1, 15 times 0, 1, 16 times 0, 1, 17 times 0, 1, 17 times 0, 1, 17 times 0, 1, 17 times 0, 1, 18 times 0, 1, 19 times 0, 1, 19 times 0, 1, 20 times 0, 1, 20 times 0, 1, 21 times 0, 1, 21 times 0, 1, 21 times 0, 1, 22 times 0, 1, 23 times 0, 1, 24 times 0, 1, 24 times 0, 1, 24 times 0, 1, 25 times 0, 1, 26 times 0, 1, 27 times 0, 1, 27 times 0, 1, 28 times 0, 1, 28 times 0, 1, 29 times 0, 1, 30 times 0, 1, 31 times 0, 1, 32 times 0, 1, 32 times 0, 1, 33 times 0, 1, 34 times 0, 1, 34 times 0, 1, 36 times 0, 1, 36 times 0, 1, 38 times 0, 1, 38 times 0, 1, 39 times 0, 1, 40 times 0, 1, 41 times 0, 1, 43 times 0, 1, 43 times 0, 1, 44 times 0, 1, 45 times 0, 1, 47 times 0, 1, 48 times 0, 1, 48 times 0, 1, 50 times 0, 1, 51 times 0, 1, 53 times 0, 1, 53 times 0, 1, 55 times 0, 1, 57 times 0, 1, 57 times 0, 1, 59 times 0, 1, 61 times 0, 1, 62 times 0, 1, 63 times 0, 1, 65 times 0, 1, 67 times 0) [i] based on linear OA(461, 62, F4, 61) (dual of [62, 1, 62]-code or 62-arc in PG(60,4)), using
- dual of repetition code with length 62 [i]
- 2771 step Varšamov–Edel lengthening with (ri) = (17, 7, 4, 3, 2, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 9 times 0, 1, 9 times 0, 1, 9 times 0, 1, 10 times 0, 1, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 11 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 12 times 0, 1, 13 times 0, 1, 13 times 0, 1, 13 times 0, 1, 14 times 0, 1, 14 times 0, 1, 14 times 0, 1, 15 times 0, 1, 15 times 0, 1, 15 times 0, 1, 16 times 0, 1, 17 times 0, 1, 17 times 0, 1, 17 times 0, 1, 17 times 0, 1, 18 times 0, 1, 19 times 0, 1, 19 times 0, 1, 20 times 0, 1, 20 times 0, 1, 21 times 0, 1, 21 times 0, 1, 21 times 0, 1, 22 times 0, 1, 23 times 0, 1, 24 times 0, 1, 24 times 0, 1, 24 times 0, 1, 25 times 0, 1, 26 times 0, 1, 27 times 0, 1, 27 times 0, 1, 28 times 0, 1, 28 times 0, 1, 29 times 0, 1, 30 times 0, 1, 31 times 0, 1, 32 times 0, 1, 32 times 0, 1, 33 times 0, 1, 34 times 0, 1, 34 times 0, 1, 36 times 0, 1, 36 times 0, 1, 38 times 0, 1, 38 times 0, 1, 39 times 0, 1, 40 times 0, 1, 41 times 0, 1, 43 times 0, 1, 43 times 0, 1, 44 times 0, 1, 45 times 0, 1, 47 times 0, 1, 48 times 0, 1, 48 times 0, 1, 50 times 0, 1, 51 times 0, 1, 53 times 0, 1, 53 times 0, 1, 55 times 0, 1, 57 times 0, 1, 57 times 0, 1, 59 times 0, 1, 61 times 0, 1, 62 times 0, 1, 63 times 0, 1, 65 times 0, 1, 67 times 0) [i] based on linear OA(461, 62, F4, 61) (dual of [62, 1, 62]-code or 62-arc in PG(60,4)), using
(258−61, 258, 577160)-Net in Base 4 — Upper bound on s
There is no (197, 258, 577161)-net in base 4, because
- 1 times m-reduction [i] would yield (197, 257, 577161)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 53633 967591 307901 758048 559831 582432 367311 440928 908877 106770 540148 105013 528236 952338 065432 833117 306787 445796 685605 205636 896358 186250 185149 948257 237297 398080 > 4257 [i]