Best Known (136−31, 136, s)-Nets in Base 4
(136−31, 136, 1040)-Net over F4 — Constructive and digital
Digital (105, 136, 1040)-net over F4, using
- trace code for nets [i] based on digital (3, 34, 260)-net over F256, using
- net from sequence [i] based on digital (3, 259)-sequence over F256, using
(136−31, 136, 2168)-Net over F4 — Digital
Digital (105, 136, 2168)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4136, 2168, F4, 31) (dual of [2168, 2032, 32]-code), using
- 2031 step Varšamov–Edel lengthening with (ri) = (9, 4, 2, 2, 2, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 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, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 10 times 0, 1, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 13 times 0, 1, 14 times 0, 1, 15 times 0, 1, 15 times 0, 1, 17 times 0, 1, 17 times 0, 1, 18 times 0, 1, 19 times 0, 1, 20 times 0, 1, 22 times 0, 1, 22 times 0, 1, 24 times 0, 1, 24 times 0, 1, 26 times 0, 1, 27 times 0, 1, 29 times 0, 1, 30 times 0, 1, 32 times 0, 1, 33 times 0, 1, 35 times 0, 1, 36 times 0, 1, 39 times 0, 1, 40 times 0, 1, 42 times 0, 1, 45 times 0, 1, 46 times 0, 1, 49 times 0, 1, 51 times 0, 1, 54 times 0, 1, 57 times 0, 1, 59 times 0, 1, 62 times 0, 1, 65 times 0, 1, 69 times 0, 1, 71 times 0, 1, 75 times 0, 1, 79 times 0, 1, 83 times 0, 1, 87 times 0, 1, 91 times 0, 1, 95 times 0) [i] based on linear OA(431, 32, F4, 31) (dual of [32, 1, 32]-code or 32-arc in PG(30,4)), using
- dual of repetition code with length 32 [i]
- 2031 step Varšamov–Edel lengthening with (ri) = (9, 4, 2, 2, 2, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 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, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 10 times 0, 1, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 13 times 0, 1, 14 times 0, 1, 15 times 0, 1, 15 times 0, 1, 17 times 0, 1, 17 times 0, 1, 18 times 0, 1, 19 times 0, 1, 20 times 0, 1, 22 times 0, 1, 22 times 0, 1, 24 times 0, 1, 24 times 0, 1, 26 times 0, 1, 27 times 0, 1, 29 times 0, 1, 30 times 0, 1, 32 times 0, 1, 33 times 0, 1, 35 times 0, 1, 36 times 0, 1, 39 times 0, 1, 40 times 0, 1, 42 times 0, 1, 45 times 0, 1, 46 times 0, 1, 49 times 0, 1, 51 times 0, 1, 54 times 0, 1, 57 times 0, 1, 59 times 0, 1, 62 times 0, 1, 65 times 0, 1, 69 times 0, 1, 71 times 0, 1, 75 times 0, 1, 79 times 0, 1, 83 times 0, 1, 87 times 0, 1, 91 times 0, 1, 95 times 0) [i] based on linear OA(431, 32, F4, 31) (dual of [32, 1, 32]-code or 32-arc in PG(30,4)), using
(136−31, 136, 561275)-Net in Base 4 — Upper bound on s
There is no (105, 136, 561276)-net in base 4, because
- 1 times m-reduction [i] would yield (105, 135, 561276)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 1897 167031 457696 869578 387058 312823 502083 193761 376758 352515 203935 218144 433471 757444 > 4135 [i]