Best Known (81, 81+20, s)-Nets in Base 2
(81, 81+20, 196)-Net over F2 — Constructive and digital
Digital (81, 101, 196)-net over F2, using
- 21 times duplication [i] based on digital (80, 100, 196)-net over F2, using
- trace code for nets [i] based on digital (5, 25, 49)-net over F16, using
- net from sequence [i] based on digital (5, 48)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 5 and N(F) ≥ 49, using
- net from sequence [i] based on digital (5, 48)-sequence over F16, using
- trace code for nets [i] based on digital (5, 25, 49)-net over F16, using
(81, 81+20, 383)-Net over F2 — Digital
Digital (81, 101, 383)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2101, 383, F2, 2, 20) (dual of [(383, 2), 665, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2101, 517, F2, 2, 20) (dual of [(517, 2), 933, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2101, 1034, F2, 20) (dual of [1034, 933, 21]-code), using
- 1 times truncation [i] based on linear OA(2102, 1035, F2, 21) (dual of [1035, 933, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(2101, 1024, F2, 21) (dual of [1024, 923, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(291, 1024, F2, 19) (dual of [1024, 933, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(21, 11, F2, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- 1 times truncation [i] based on linear OA(2102, 1035, F2, 21) (dual of [1035, 933, 22]-code), using
- OOA 2-folding [i] based on linear OA(2101, 1034, F2, 20) (dual of [1034, 933, 21]-code), using
- discarding factors / shortening the dual code based on linear OOA(2101, 517, F2, 2, 20) (dual of [(517, 2), 933, 21]-NRT-code), using
(81, 81+20, 4955)-Net in Base 2 — Upper bound on s
There is no (81, 101, 4956)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 2 536420 344083 891286 609647 703093 > 2101 [i]