Best Known (217−45, 217, s)-Nets in Base 2
(217−45, 217, 260)-Net over F2 — Constructive and digital
Digital (172, 217, 260)-net over F2, using
- t-expansion [i] based on digital (171, 217, 260)-net over F2, using
- 3 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
- 3 times m-reduction [i] based on digital (171, 220, 260)-net over F2, using
(217−45, 217, 516)-Net over F2 — Digital
Digital (172, 217, 516)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2217, 516, F2, 2, 45) (dual of [(516, 2), 815, 46]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2217, 517, F2, 2, 45) (dual of [(517, 2), 817, 46]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2217, 1034, F2, 45) (dual of [1034, 817, 46]-code), using
- discarding factors / shortening the dual code based on linear OA(2217, 1035, F2, 45) (dual of [1035, 818, 46]-code), using
- construction X applied to Ce(44) ⊂ Ce(42) [i] based on
- linear OA(2216, 1024, F2, 45) (dual of [1024, 808, 46]-code), using an extension Ce(44) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,44], and designed minimum distance d ≥ |I|+1 = 45 [i]
- linear OA(2206, 1024, F2, 43) (dual of [1024, 818, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [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(44) ⊂ Ce(42) [i] based on
- discarding factors / shortening the dual code based on linear OA(2217, 1035, F2, 45) (dual of [1035, 818, 46]-code), using
- OOA 2-folding [i] based on linear OA(2217, 1034, F2, 45) (dual of [1034, 817, 46]-code), using
- discarding factors / shortening the dual code based on linear OOA(2217, 517, F2, 2, 45) (dual of [(517, 2), 817, 46]-NRT-code), using
(217−45, 217, 8141)-Net in Base 2 — Upper bound on s
There is no (172, 217, 8142)-net in base 2, because
- 1 times m-reduction [i] would yield (172, 216, 8142)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 105514 219743 574369 576849 681495 011717 992016 319796 204566 407247 959216 > 2216 [i]