Best Known (75, 75+18, s)-Nets in Base 2
(75, 75+18, 196)-Net over F2 — Constructive and digital
Digital (75, 93, 196)-net over F2, using
- 21 times duplication [i] based on digital (74, 92, 196)-net over F2, using
- trace code for nets [i] based on digital (5, 23, 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, 23, 49)-net over F16, using
(75, 75+18, 410)-Net over F2 — Digital
Digital (75, 93, 410)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(293, 410, F2, 2, 18) (dual of [(410, 2), 727, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(293, 523, F2, 2, 18) (dual of [(523, 2), 953, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(293, 1046, F2, 18) (dual of [1046, 953, 19]-code), using
- construction XX applied to C1 = C([1021,14]), C2 = C([1,16]), C3 = C1 + C2 = C([1,14]), and C∩ = C1 ∩ C2 = C([1021,16]) [i] based on
- linear OA(281, 1023, F2, 17) (dual of [1023, 942, 18]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,14}, and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(280, 1023, F2, 16) (dual of [1023, 943, 17]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(291, 1023, F2, 19) (dual of [1023, 932, 20]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,16}, and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(270, 1023, F2, 14) (dual of [1023, 953, 15]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(21, 12, F2, 1) (dual of [12, 11, 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
- 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 (see above)
- construction XX applied to C1 = C([1021,14]), C2 = C([1,16]), C3 = C1 + C2 = C([1,14]), and C∩ = C1 ∩ C2 = C([1021,16]) [i] based on
- OOA 2-folding [i] based on linear OA(293, 1046, F2, 18) (dual of [1046, 953, 19]-code), using
- discarding factors / shortening the dual code based on linear OOA(293, 523, F2, 2, 18) (dual of [(523, 2), 953, 19]-NRT-code), using
(75, 75+18, 5337)-Net in Base 2 — Upper bound on s
There is no (75, 93, 5338)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 9911 542672 867876 619003 321461 > 293 [i]