Best Known (65, 89, s)-Nets in Base 7
(65, 89, 344)-Net over F7 — Constructive and digital
Digital (65, 89, 344)-net over F7, using
- base reduction for projective spaces (embedding PG(44,49) in PG(88,7)) for nets [i] based on digital (21, 45, 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
(65, 89, 2946)-Net over F7 — Digital
Digital (65, 89, 2946)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(789, 2946, F7, 24) (dual of [2946, 2857, 25]-code), using
- 533 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 17 times 0, 1, 42 times 0, 1, 91 times 0, 1, 157 times 0, 1, 213 times 0) [i] based on linear OA(781, 2405, F7, 24) (dual of [2405, 2324, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(22) [i] based on
- linear OA(781, 2401, F7, 24) (dual of [2401, 2320, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(777, 2401, F7, 23) (dual of [2401, 2324, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(70, 4, F7, 0) (dual of [4, 4, 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(23) ⊂ Ce(22) [i] based on
- 533 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 17 times 0, 1, 42 times 0, 1, 91 times 0, 1, 157 times 0, 1, 213 times 0) [i] based on linear OA(781, 2405, F7, 24) (dual of [2405, 2324, 25]-code), using
(65, 89, 1633122)-Net in Base 7 — Upper bound on s
There is no (65, 89, 1633123)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 1635 794398 134308 040171 066290 302546 188478 413145 149568 354855 895225 474561 462233 > 789 [i]