Best Known (131−36, 131, s)-Nets in Base 8
(131−36, 131, 1026)-Net over F8 — Constructive and digital
Digital (95, 131, 1026)-net over F8, using
- 3 times m-reduction [i] based on digital (95, 134, 1026)-net over F8, using
- trace code for nets [i] based on digital (28, 67, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- trace code for nets [i] based on digital (28, 67, 513)-net over F64, using
(131−36, 131, 4787)-Net over F8 — Digital
Digital (95, 131, 4787)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8131, 4787, F8, 36) (dual of [4787, 4656, 37]-code), using
- 681 step Varšamov–Edel lengthening with (ri) = (2, 6 times 0, 1, 43 times 0, 1, 138 times 0, 1, 223 times 0, 1, 266 times 0) [i] based on linear OA(8125, 4100, F8, 36) (dual of [4100, 3975, 37]-code), using
- construction X applied to Ce(35) ⊂ Ce(34) [i] based on
- linear OA(8125, 4096, F8, 36) (dual of [4096, 3971, 37]-code), using an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(8121, 4096, F8, 35) (dual of [4096, 3975, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(80, 4, F8, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(35) ⊂ Ce(34) [i] based on
- 681 step Varšamov–Edel lengthening with (ri) = (2, 6 times 0, 1, 43 times 0, 1, 138 times 0, 1, 223 times 0, 1, 266 times 0) [i] based on linear OA(8125, 4100, F8, 36) (dual of [4100, 3975, 37]-code), using
(131−36, 131, 4031986)-Net in Base 8 — Upper bound on s
There is no (95, 131, 4031987)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 20173 877654 910944 531842 851360 909991 624586 601736 406280 115059 401952 932764 712304 650413 762955 408825 795636 739715 401125 780131 > 8131 [i]