Best Known (44−17, 44, s)-Nets in Base 8
(44−17, 44, 256)-Net over F8 — Constructive and digital
Digital (27, 44, 256)-net over F8, using
- trace code for nets [i] based on digital (5, 22, 128)-net over F64, using
- net from sequence [i] based on digital (5, 127)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 5 and N(F) ≥ 128, using
- net from sequence [i] based on digital (5, 127)-sequence over F64, using
(44−17, 44, 300)-Net in Base 8 — Constructive
(27, 44, 300)-net in base 8, using
- 82 times duplication [i] based on (25, 42, 300)-net in base 8, using
- trace code for nets [i] based on (4, 21, 150)-net in base 64, using
- base change [i] based on digital (1, 18, 150)-net over F128, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 1 and N(F) ≥ 150, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- base change [i] based on digital (1, 18, 150)-net over F128, using
- trace code for nets [i] based on (4, 21, 150)-net in base 64, using
(44−17, 44, 348)-Net over F8 — Digital
Digital (27, 44, 348)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(844, 348, F8, 17) (dual of [348, 304, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(844, 516, F8, 17) (dual of [516, 472, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(843, 512, F8, 17) (dual of [512, 469, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(840, 512, F8, 15) (dual of [512, 472, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(81, 4, F8, 1) (dual of [4, 3, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(844, 516, F8, 17) (dual of [516, 472, 18]-code), using
(44−17, 44, 38428)-Net in Base 8 — Upper bound on s
There is no (27, 44, 38429)-net in base 8, because
- 1 times m-reduction [i] would yield (27, 43, 38429)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 680 700674 667411 621956 249988 333503 282825 > 843 [i]