Best Known (103−29, 103, s)-Nets in Base 7
(103−29, 103, 688)-Net over F7 — Constructive and digital
Digital (74, 103, 688)-net over F7, using
- 3 times m-reduction [i] based on digital (74, 106, 688)-net over F7, using
- trace code for nets [i] based on digital (21, 53, 344)-net over F49, using
- net from sequence [i] based on digital (21, 343)-sequence over F49, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 21 and N(F) ≥ 344, using
- the Hermitian function field over F49 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 21 and N(F) ≥ 344, using
- net from sequence [i] based on digital (21, 343)-sequence over F49, using
- trace code for nets [i] based on digital (21, 53, 344)-net over F49, using
(103−29, 103, 2519)-Net over F7 — Digital
Digital (74, 103, 2519)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7103, 2519, F7, 29) (dual of [2519, 2416, 30]-code), using
- 111 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 0, 1, 11 times 0, 1, 28 times 0, 1, 64 times 0) [i] based on linear OA(797, 2402, F7, 29) (dual of [2402, 2305, 30]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2402 | 78−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- 111 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 0, 1, 11 times 0, 1, 28 times 0, 1, 64 times 0) [i] based on linear OA(797, 2402, F7, 29) (dual of [2402, 2305, 30]-code), using
(103−29, 103, 1446927)-Net in Base 7 — Upper bound on s
There is no (74, 103, 1446928)-net in base 7, because
- 1 times m-reduction [i] would yield (74, 102, 1446928)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 158 490473 994125 185082 726583 862380 881145 550038 151078 442662 757276 471165 108101 636197 327841 > 7102 [i]