Best Known (38, 38+23, s)-Nets in Base 7
(38, 38+23, 116)-Net over F7 — Constructive and digital
Digital (38, 61, 116)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (4, 15, 16)-net over F7, using
- 1 times m-reduction [i] based on digital (4, 16, 16)-net over F7, using
- digital (23, 46, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 23, 50)-net over F49, using
- net from sequence [i] based on digital (0, 49)-sequence over F49, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 0 and N(F) ≥ 50, using
- the rational function field F49(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 49)-sequence over F49, using
- trace code for nets [i] based on digital (0, 23, 50)-net over F49, using
- digital (4, 15, 16)-net over F7, using
(38, 38+23, 362)-Net over F7 — Digital
Digital (38, 61, 362)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(761, 362, F7, 23) (dual of [362, 301, 24]-code), using
- 13 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 9 times 0) [i] based on linear OA(758, 346, F7, 23) (dual of [346, 288, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(758, 343, F7, 23) (dual of [343, 285, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(755, 343, F7, 22) (dual of [343, 288, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(70, 3, F7, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- 13 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 9 times 0) [i] based on linear OA(758, 346, F7, 23) (dual of [346, 288, 24]-code), using
(38, 38+23, 33296)-Net in Base 7 — Upper bound on s
There is no (38, 61, 33297)-net in base 7, because
- 1 times m-reduction [i] would yield (38, 60, 33297)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 508 094434 671831 383160 446902 076488 976269 782291 850903 > 760 [i]