Best Known (42, 42+10, s)-Nets in Base 16
(42, 42+10, 209740)-Net over F16 — Constructive and digital
Digital (42, 52, 209740)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (1, 6, 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 (36, 46, 209716)-net over F16, using
- net defined by OOA [i] based on linear OOA(1646, 209716, F16, 10, 10) (dual of [(209716, 10), 2097114, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(1646, 1048580, F16, 10) (dual of [1048580, 1048534, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(1646, 1048581, F16, 10) (dual of [1048581, 1048535, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(1646, 1048576, F16, 10) (dual of [1048576, 1048530, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(1641, 1048576, F16, 9) (dual of [1048576, 1048535, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(160, 5, F16, 0) (dual of [5, 5, 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(9) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(1646, 1048581, F16, 10) (dual of [1048581, 1048535, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(1646, 1048580, F16, 10) (dual of [1048580, 1048534, 11]-code), using
- net defined by OOA [i] based on linear OOA(1646, 209716, F16, 10, 10) (dual of [(209716, 10), 2097114, 11]-NRT-code), using
- digital (1, 6, 24)-net over F16, using
(42, 42+10, 419432)-Net in Base 16 — Constructive
(42, 52, 419432)-net in base 16, using
- net defined by OOA [i] based on OOA(1652, 419432, S16, 10, 10), using
- OA 5-folding and stacking [i] based on OA(1652, 2097160, S16, 10), using
- 1 times code embedding in larger space [i] based on OA(1651, 2097159, S16, 10), using
- discarding parts of the base [i] based on linear OA(12829, 2097159, F128, 10) (dual of [2097159, 2097130, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(12828, 2097152, F128, 10) (dual of [2097152, 2097124, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(12822, 2097152, F128, 8) (dual of [2097152, 2097130, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(1281, 7, F128, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- discarding parts of the base [i] based on linear OA(12829, 2097159, F128, 10) (dual of [2097159, 2097130, 11]-code), using
- 1 times code embedding in larger space [i] based on OA(1651, 2097159, S16, 10), using
- OA 5-folding and stacking [i] based on OA(1652, 2097160, S16, 10), using
(42, 42+10, 2504948)-Net over F16 — Digital
Digital (42, 52, 2504948)-net over F16, using
(42, 42+10, large)-Net in Base 16 — Upper bound on s
There is no (42, 52, large)-net in base 16, because
- 8 times m-reduction [i] would yield (42, 44, large)-net in base 16, but