Best Known (38−21, 38, s)-Nets in Base 16
(38−21, 38, 89)-Net over F16 — Constructive and digital
Digital (17, 38, 89)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (1, 11, 24)-net over F16, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 1 and N(F) ≥ 24, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- digital (6, 27, 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
- digital (1, 11, 24)-net over F16, using
(38−21, 38, 120)-Net over F16 — Digital
Digital (17, 38, 120)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(1638, 120, F16, 2, 21) (dual of [(120, 2), 202, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1638, 129, F16, 2, 21) (dual of [(129, 2), 220, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(1638, 258, F16, 21) (dual of [258, 220, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- linear OA(1638, 256, F16, 21) (dual of [256, 218, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 255 = 162−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(1636, 256, F16, 20) (dual of [256, 220, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 255 = 162−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(160, 2, F16, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- OOA 2-folding [i] based on linear OA(1638, 258, F16, 21) (dual of [258, 220, 22]-code), using
- discarding factors / shortening the dual code based on linear OOA(1638, 129, F16, 2, 21) (dual of [(129, 2), 220, 22]-NRT-code), using
(38−21, 38, 129)-Net in Base 16 — Constructive
(17, 38, 129)-net in base 16, using
- 1 times m-reduction [i] based on (17, 39, 129)-net in base 16, using
- base change [i] based on (4, 26, 129)-net in base 64, using
- 2 times m-reduction [i] based on (4, 28, 129)-net in base 64, using
- base change [i] based on digital (0, 24, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- base change [i] based on digital (0, 24, 129)-net over F128, using
- 2 times m-reduction [i] based on (4, 28, 129)-net in base 64, using
- base change [i] based on (4, 26, 129)-net in base 64, using
(38−21, 38, 8607)-Net in Base 16 — Upper bound on s
There is no (17, 38, 8608)-net in base 16, because
- 1 times m-reduction [i] would yield (17, 37, 8608)-net in base 16, but
- the generalized Rao bound for nets shows that 16m ≥ 357 089959 322280 975946 306396 000666 754221 055701 > 1637 [i]