Best Known (37−20, 37, s)-Nets in Base 16
(37−20, 37, 89)-Net over F16 — Constructive and digital
Digital (17, 37, 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, 26, 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
(37−20, 37, 129)-Net in Base 16 — Constructive
(17, 37, 129)-net in base 16, using
- 2 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
(37−20, 37, 131)-Net over F16 — Digital
Digital (17, 37, 131)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(1637, 131, F16, 2, 20) (dual of [(131, 2), 225, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(1637, 262, F16, 20) (dual of [262, 225, 21]-code), using
- construction X applied to C([119,138]) ⊂ C([120,137]) [i] based on
- linear OA(1636, 257, F16, 20) (dual of [257, 221, 21]-code), using the BCH-code C(I) with length 257 | 164−1, defining interval I = {119,120,…,138}, and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(1632, 257, F16, 18) (dual of [257, 225, 19]-code), using the BCH-code C(I) with length 257 | 164−1, defining interval I = {120,121,…,137}, and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(161, 5, F16, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(161, s, F16, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([119,138]) ⊂ C([120,137]) [i] based on
- OOA 2-folding [i] based on linear OA(1637, 262, F16, 20) (dual of [262, 225, 21]-code), using
(37−20, 37, 8607)-Net in Base 16 — Upper bound on s
There is no (17, 37, 8608)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 357 089959 322280 975946 306396 000666 754221 055701 > 1637 [i]