Best Known (112−22, 112, s)-Nets in Base 2
(112−22, 112, 260)-Net over F2 — Constructive and digital
Digital (90, 112, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 28, 65)-net over F16, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- the Hermitian function field over F16 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
(112−22, 112, 412)-Net over F2 — Digital
Digital (90, 112, 412)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2112, 412, F2, 2, 22) (dual of [(412, 2), 712, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2112, 522, F2, 2, 22) (dual of [(522, 2), 932, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2112, 1044, F2, 22) (dual of [1044, 932, 23]-code), using
- 1 times truncation [i] based on linear OA(2113, 1045, F2, 23) (dual of [1045, 932, 24]-code), using
- construction XX applied to C1 = C([1021,18]), C2 = C([0,20]), C3 = C1 + C2 = C([0,18]), and C∩ = C1 ∩ C2 = C([1021,20]) [i] based on
- linear OA(2101, 1023, F2, 21) (dual of [1023, 922, 22]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,18}, and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(2101, 1023, F2, 21) (dual of [1023, 922, 22]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(2111, 1023, F2, 23) (dual of [1023, 912, 24]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,20}, and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(291, 1023, F2, 19) (dual of [1023, 932, 20]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [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
- linear OA(21, 11, F2, 1) (dual of [11, 10, 2]-code) (see above)
- construction XX applied to C1 = C([1021,18]), C2 = C([0,20]), C3 = C1 + C2 = C([0,18]), and C∩ = C1 ∩ C2 = C([1021,20]) [i] based on
- 1 times truncation [i] based on linear OA(2113, 1045, F2, 23) (dual of [1045, 932, 24]-code), using
- OOA 2-folding [i] based on linear OA(2112, 1044, F2, 22) (dual of [1044, 932, 23]-code), using
- discarding factors / shortening the dual code based on linear OOA(2112, 522, F2, 2, 22) (dual of [(522, 2), 932, 23]-NRT-code), using
(112−22, 112, 5686)-Net in Base 2 — Upper bound on s
There is no (90, 112, 5687)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 5199 534967 706104 963141 564727 023284 > 2112 [i]