Best Known (253−59, 253, s)-Nets in Base 4
(253−59, 253, 1044)-Net over F4 — Constructive and digital
Digital (194, 253, 1044)-net over F4, using
- 41 times duplication [i] based on digital (193, 252, 1044)-net over F4, using
- trace code for nets [i] based on digital (4, 63, 261)-net over F256, using
- net from sequence [i] based on digital (4, 260)-sequence over F256, using
- trace code for nets [i] based on digital (4, 63, 261)-net over F256, using
(253−59, 253, 3194)-Net over F4 — Digital
Digital (194, 253, 3194)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4253, 3194, F4, 59) (dual of [3194, 2941, 60]-code), using
- 2940 step Varšamov–Edel lengthening with (ri) = (16, 7, 4, 3, 2, 2, 2, 1, 1, 1, 2, 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, 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, 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, 5 times 0, 1, 4 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, 6 times 0, 1, 7 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 7 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, 10 times 0, 1, 10 times 0, 1, 10 times 0, 1, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 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, 15 times 0, 1, 15 times 0, 1, 15 times 0, 1, 16 times 0, 1, 16 times 0, 1, 17 times 0, 1, 17 times 0, 1, 18 times 0, 1, 18 times 0, 1, 19 times 0, 1, 19 times 0, 1, 19 times 0, 1, 20 times 0, 1, 21 times 0, 1, 21 times 0, 1, 22 times 0, 1, 22 times 0, 1, 23 times 0, 1, 23 times 0, 1, 24 times 0, 1, 25 times 0, 1, 26 times 0, 1, 26 times 0, 1, 27 times 0, 1, 27 times 0, 1, 28 times 0, 1, 29 times 0, 1, 30 times 0, 1, 30 times 0, 1, 31 times 0, 1, 32 times 0, 1, 33 times 0, 1, 33 times 0, 1, 35 times 0, 1, 35 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, 42 times 0, 1, 44 times 0, 1, 44 times 0, 1, 45 times 0, 1, 47 times 0, 1, 48 times 0, 1, 49 times 0, 1, 50 times 0, 1, 52 times 0, 1, 52 times 0, 1, 54 times 0, 1, 56 times 0, 1, 57 times 0, 1, 58 times 0, 1, 60 times 0, 1, 61 times 0, 1, 63 times 0, 1, 64 times 0, 1, 66 times 0, 1, 68 times 0, 1, 69 times 0, 1, 71 times 0, 1, 73 times 0) [i] based on linear OA(459, 60, F4, 59) (dual of [60, 1, 60]-code or 60-arc in PG(58,4)), using
- dual of repetition code with length 60 [i]
- 2940 step Varšamov–Edel lengthening with (ri) = (16, 7, 4, 3, 2, 2, 2, 1, 1, 1, 2, 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, 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, 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, 5 times 0, 1, 4 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, 6 times 0, 1, 7 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 7 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, 10 times 0, 1, 10 times 0, 1, 10 times 0, 1, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 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, 15 times 0, 1, 15 times 0, 1, 15 times 0, 1, 16 times 0, 1, 16 times 0, 1, 17 times 0, 1, 17 times 0, 1, 18 times 0, 1, 18 times 0, 1, 19 times 0, 1, 19 times 0, 1, 19 times 0, 1, 20 times 0, 1, 21 times 0, 1, 21 times 0, 1, 22 times 0, 1, 22 times 0, 1, 23 times 0, 1, 23 times 0, 1, 24 times 0, 1, 25 times 0, 1, 26 times 0, 1, 26 times 0, 1, 27 times 0, 1, 27 times 0, 1, 28 times 0, 1, 29 times 0, 1, 30 times 0, 1, 30 times 0, 1, 31 times 0, 1, 32 times 0, 1, 33 times 0, 1, 33 times 0, 1, 35 times 0, 1, 35 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, 42 times 0, 1, 44 times 0, 1, 44 times 0, 1, 45 times 0, 1, 47 times 0, 1, 48 times 0, 1, 49 times 0, 1, 50 times 0, 1, 52 times 0, 1, 52 times 0, 1, 54 times 0, 1, 56 times 0, 1, 57 times 0, 1, 58 times 0, 1, 60 times 0, 1, 61 times 0, 1, 63 times 0, 1, 64 times 0, 1, 66 times 0, 1, 68 times 0, 1, 69 times 0, 1, 71 times 0, 1, 73 times 0) [i] based on linear OA(459, 60, F4, 59) (dual of [60, 1, 60]-code or 60-arc in PG(58,4)), using
(253−59, 253, 663252)-Net in Base 4 — Upper bound on s
There is no (194, 253, 663253)-net in base 4, because
- 1 times m-reduction [i] would yield (194, 252, 663253)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 52 375047 286378 114073 873691 046308 241784 351660 877735 935869 718643 588907 708014 696426 223695 215890 396639 109787 527632 373010 198847 835500 305315 557803 536299 260384 > 4252 [i]