Best Known (90, 90+34, s)-Nets in Base 8
(90, 90+34, 1026)-Net over F8 — Constructive and digital
Digital (90, 124, 1026)-net over F8, using
- trace code for nets [i] based on digital (28, 62, 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
(90, 90+34, 4679)-Net over F8 — Digital
Digital (90, 124, 4679)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8124, 4679, F8, 34) (dual of [4679, 4555, 35]-code), using
- 572 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 13 times 0, 1, 37 times 0, 1, 91 times 0, 1, 174 times 0, 1, 248 times 0) [i] based on linear OA(8117, 4100, F8, 34) (dual of [4100, 3983, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(32) [i] based on
- linear OA(8117, 4096, F8, 34) (dual of [4096, 3979, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(8113, 4096, F8, 33) (dual of [4096, 3983, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [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(33) ⊂ Ce(32) [i] based on
- 572 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 13 times 0, 1, 37 times 0, 1, 91 times 0, 1, 174 times 0, 1, 248 times 0) [i] based on linear OA(8117, 4100, F8, 34) (dual of [4100, 3983, 35]-code), using
(90, 90+34, 3963607)-Net in Base 8 — Upper bound on s
There is no (90, 124, 3963608)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 9619 653465 961955 001037 263503 787990 592344 770773 477606 747335 894191 034786 685801 386186 554489 002514 973348 487562 952577 > 8124 [i]