Best Known (78, 78+37, s)-Nets in Base 5
(78, 78+37, 296)-Net over F5 — Constructive and digital
Digital (78, 115, 296)-net over F5, using
- 3 times m-reduction [i] based on digital (78, 118, 296)-net over F5, using
- trace code for nets [i] based on digital (19, 59, 148)-net over F25, using
- net from sequence [i] based on digital (19, 147)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 19 and N(F) ≥ 148, using
- net from sequence [i] based on digital (19, 147)-sequence over F25, using
- trace code for nets [i] based on digital (19, 59, 148)-net over F25, using
(78, 78+37, 629)-Net over F5 — Digital
Digital (78, 115, 629)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5115, 629, F5, 37) (dual of [629, 514, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(35) [i] based on
- linear OA(5115, 625, F5, 37) (dual of [625, 510, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(5111, 625, F5, 36) (dual of [625, 514, 37]-code), using an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(50, 4, F5, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(36) ⊂ Ce(35) [i] based on
(78, 78+37, 50439)-Net in Base 5 — Upper bound on s
There is no (78, 115, 50440)-net in base 5, because
- 1 times m-reduction [i] would yield (78, 114, 50440)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 48 162003 657893 349595 163484 707676 091131 439724 492168 452536 022454 894364 029740 274305 > 5114 [i]