Best Known (56, 83, s)-Nets in Base 5
(56, 83, 252)-Net over F5 — Constructive and digital
Digital (56, 83, 252)-net over F5, using
- 9 times m-reduction [i] based on digital (56, 92, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 46, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- the Hermitian function field over F25 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- trace code for nets [i] based on digital (10, 46, 126)-net over F25, using
(56, 83, 483)-Net over F5 — Digital
Digital (56, 83, 483)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(583, 483, F5, 27) (dual of [483, 400, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(583, 624, F5, 27) (dual of [624, 541, 28]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,26], and designed minimum distance d ≥ |I|+1 = 28 [i]
- discarding factors / shortening the dual code based on linear OA(583, 624, F5, 27) (dual of [624, 541, 28]-code), using
(56, 83, 36318)-Net in Base 5 — Upper bound on s
There is no (56, 83, 36319)-net in base 5, because
- 1 times m-reduction [i] would yield (56, 82, 36319)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 2068 605358 238350 888483 102994 741877 345828 823511 850523 529037 > 582 [i]