Best Known (130−24, 130, s)-Nets in Base 16
(130−24, 130, 87447)-Net over F16 — Constructive and digital
Digital (106, 130, 87447)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (6, 18, 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 (88, 112, 87382)-net over F16, using
- net defined by OOA [i] based on linear OOA(16112, 87382, F16, 24, 24) (dual of [(87382, 24), 2097056, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(16112, 1048584, F16, 24) (dual of [1048584, 1048472, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(16112, 1048587, F16, 24) (dual of [1048587, 1048475, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(21) [i] based on
- linear OA(16111, 1048576, F16, 24) (dual of [1048576, 1048465, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(16101, 1048576, F16, 22) (dual of [1048576, 1048475, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(161, 11, F16, 1) (dual of [11, 10, 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 Ce(23) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(16112, 1048587, F16, 24) (dual of [1048587, 1048475, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(16112, 1048584, F16, 24) (dual of [1048584, 1048472, 25]-code), using
- net defined by OOA [i] based on linear OOA(16112, 87382, F16, 24, 24) (dual of [(87382, 24), 2097056, 25]-NRT-code), using
- digital (6, 18, 65)-net over F16, using
(130−24, 130, 174764)-Net in Base 16 — Constructive
(106, 130, 174764)-net in base 16, using
- 161 times duplication [i] based on (105, 129, 174764)-net in base 16, using
- net defined by OOA [i] based on OOA(16129, 174764, S16, 24, 24), using
- OA 12-folding and stacking [i] based on OA(16129, 2097168, S16, 24), using
- 1 times code embedding in larger space [i] based on OA(16128, 2097167, S16, 24), using
- discarding parts of the base [i] based on linear OA(12873, 2097167, F128, 24) (dual of [2097167, 2097094, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(19) [i] based on
- linear OA(12870, 2097152, F128, 24) (dual of [2097152, 2097082, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(12858, 2097152, F128, 20) (dual of [2097152, 2097094, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(1283, 15, F128, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,128) or 15-cap in PG(2,128)), using
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- Reed–Solomon code RS(125,128) [i]
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- construction X applied to Ce(23) ⊂ Ce(19) [i] based on
- discarding parts of the base [i] based on linear OA(12873, 2097167, F128, 24) (dual of [2097167, 2097094, 25]-code), using
- 1 times code embedding in larger space [i] based on OA(16128, 2097167, S16, 24), using
- OA 12-folding and stacking [i] based on OA(16129, 2097168, S16, 24), using
- net defined by OOA [i] based on OOA(16129, 174764, S16, 24, 24), using
(130−24, 130, 4020331)-Net over F16 — Digital
Digital (106, 130, 4020331)-net over F16, using
(130−24, 130, large)-Net in Base 16 — Upper bound on s
There is no (106, 130, large)-net in base 16, because
- 22 times m-reduction [i] would yield (106, 108, large)-net in base 16, but