Best Known (218−45, 218, s)-Nets in Base 2
(218−45, 218, 260)-Net over F2 — Constructive and digital
Digital (173, 218, 260)-net over F2, using
- t-expansion [i] based on digital (171, 218, 260)-net over F2, using
- 2 times m-reduction [i] based on digital (171, 220, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 55, 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
- trace code for nets [i] based on digital (6, 55, 65)-net over F16, using
- 2 times m-reduction [i] based on digital (171, 220, 260)-net over F2, using
(218−45, 218, 522)-Net over F2 — Digital
Digital (173, 218, 522)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2218, 522, F2, 2, 45) (dual of [(522, 2), 826, 46]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2218, 1044, F2, 45) (dual of [1044, 826, 46]-code), using
- discarding factors / shortening the dual code based on linear OA(2218, 1045, F2, 45) (dual of [1045, 827, 46]-code), using
- construction XX applied to C1 = C([1021,40]), C2 = C([0,42]), C3 = C1 + C2 = C([0,40]), and C∩ = C1 ∩ C2 = C([1021,42]) [i] based on
- linear OA(2206, 1023, F2, 43) (dual of [1023, 817, 44]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,40}, and designed minimum distance d ≥ |I|+1 = 44 [i]
- linear OA(2206, 1023, F2, 43) (dual of [1023, 817, 44]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,42], and designed minimum distance d ≥ |I|+1 = 44 [i]
- linear OA(2216, 1023, F2, 45) (dual of [1023, 807, 46]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,42}, and designed minimum distance d ≥ |I|+1 = 46 [i]
- linear OA(2196, 1023, F2, 41) (dual of [1023, 827, 42]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,40], and designed minimum distance d ≥ |I|+1 = 42 [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([1021,40]), C2 = C([0,42]), C3 = C1 + C2 = C([0,40]), and C∩ = C1 ∩ C2 = C([1021,42]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2218, 1045, F2, 45) (dual of [1045, 827, 46]-code), using
- OOA 2-folding [i] based on linear OA(2218, 1044, F2, 45) (dual of [1044, 826, 46]-code), using
(218−45, 218, 8402)-Net in Base 2 — Upper bound on s
There is no (173, 218, 8403)-net in base 2, because
- 1 times m-reduction [i] would yield (173, 217, 8403)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 210672 123236 130596 727417 013126 346398 552905 186880 031237 122979 908128 > 2217 [i]