Best Known (83, 110, s)-Nets in Base 5
(83, 110, 400)-Net over F5 — Constructive and digital
Digital (83, 110, 400)-net over F5, using
- 6 times m-reduction [i] based on digital (83, 116, 400)-net over F5, using
- trace code for nets [i] based on digital (25, 58, 200)-net over F25, using
- net from sequence [i] based on digital (25, 199)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 25 and N(F) ≥ 200, using
- net from sequence [i] based on digital (25, 199)-sequence over F25, using
- trace code for nets [i] based on digital (25, 58, 200)-net over F25, using
(83, 110, 2821)-Net over F5 — Digital
Digital (83, 110, 2821)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5110, 2821, F5, 27) (dual of [2821, 2711, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(5110, 3144, F5, 27) (dual of [3144, 3034, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- linear OA(5106, 3125, F5, 27) (dual of [3125, 3019, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(591, 3125, F5, 23) (dual of [3125, 3034, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(54, 19, F5, 3) (dual of [19, 15, 4]-code or 19-cap in PG(3,5)), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(5110, 3144, F5, 27) (dual of [3144, 3034, 28]-code), using
(83, 110, 1027867)-Net in Base 5 — Upper bound on s
There is no (83, 110, 1027868)-net in base 5, because
- 1 times m-reduction [i] would yield (83, 109, 1027868)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 15407 631101 887986 708525 540558 365468 703626 744768 182463 057865 084250 005342 343665 > 5109 [i]