Best Known (18−4, 18, s)-Nets in Base 16
(18−4, 18, 524307)-Net over F16 — Constructive and digital
Digital (14, 18, 524307)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (0, 2, 17)-net over F16, using
- digital (12, 16, 524290)-net over F16, using
- net defined by OOA [i] based on linear OOA(1616, 524290, F16, 4, 4) (dual of [(524290, 4), 2097144, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(1616, 1048580, F16, 4) (dual of [1048580, 1048564, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(1616, 1048581, F16, 4) (dual of [1048581, 1048565, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(1616, 1048576, F16, 4) (dual of [1048576, 1048560, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(1611, 1048576, F16, 3) (dual of [1048576, 1048565, 4]-code or 1048576-cap in PG(10,16)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [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(3) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(1616, 1048581, F16, 4) (dual of [1048581, 1048565, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(1616, 1048580, F16, 4) (dual of [1048580, 1048564, 5]-code), using
- net defined by OOA [i] based on linear OOA(1616, 524290, F16, 4, 4) (dual of [(524290, 4), 2097144, 5]-NRT-code), using
(18−4, 18, 1048577)-Net in Base 16 — Constructive
(14, 18, 1048577)-net in base 16, using
- net defined by OOA [i] based on OOA(1618, 1048577, S16, 4, 4), using
- OA 2-folding and stacking [i] based on OA(1618, 2097154, S16, 4), using
- discarding factors based on OA(1618, 2097155, S16, 4), using
- discarding parts of the base [i] based on linear OA(12810, 2097155, F128, 4) (dual of [2097155, 2097145, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(12810, 2097152, F128, 4) (dual of [2097152, 2097142, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(1287, 2097152, F128, 3) (dual of [2097152, 2097145, 4]-code or 2097152-cap in PG(6,128)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- discarding parts of the base [i] based on linear OA(12810, 2097155, F128, 4) (dual of [2097155, 2097145, 5]-code), using
- discarding factors based on OA(1618, 2097155, S16, 4), using
- OA 2-folding and stacking [i] based on OA(1618, 2097154, S16, 4), using
(18−4, 18, 2032416)-Net over F16 — Digital
Digital (14, 18, 2032416)-net over F16, using
(18−4, 18, 2097155)-Net in Base 16
(14, 18, 2097155)-net in base 16, using
- base change [i] based on (8, 12, 2097155)-net in base 64, using
- net defined by OOA [i] based on OOA(6412, 2097155, S64, 4, 4), using
- appending kth column [i] based on OOA(6412, 2097155, S64, 3, 4), using
- discarding parts of the base [i] based on linear OOA(12810, 2097155, F128, 3, 4) (dual of [(2097155, 3), 6291455, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12810, 2097155, F128, 4) (dual of [2097155, 2097145, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(12810, 2097152, F128, 4) (dual of [2097152, 2097142, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(1287, 2097152, F128, 3) (dual of [2097152, 2097145, 4]-code or 2097152-cap in PG(6,128)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12810, 2097155, F128, 4) (dual of [2097155, 2097145, 5]-code), using
- discarding parts of the base [i] based on linear OOA(12810, 2097155, F128, 3, 4) (dual of [(2097155, 3), 6291455, 5]-NRT-code), using
- appending kth column [i] based on OOA(6412, 2097155, S64, 3, 4), using
- net defined by OOA [i] based on OOA(6412, 2097155, S64, 4, 4), using
(18−4, 18, large)-Net in Base 16 — Upper bound on s
There is no (14, 18, large)-net in base 16, because
- 2 times m-reduction [i] would yield (14, 16, large)-net in base 16, but