Best Known (72, 72+16, s)-Nets in Base 16
(72, 72+16, 131120)-Net over F16 — Constructive and digital
Digital (72, 88, 131120)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (5, 13, 49)-net over F16, using
- net from sequence [i] based on digital (5, 48)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 5 and N(F) ≥ 49, using
- net from sequence [i] based on digital (5, 48)-sequence over F16, using
- digital (59, 75, 131071)-net over F16, using
- net defined by OOA [i] based on linear OOA(1675, 131071, F16, 16, 16) (dual of [(131071, 16), 2097061, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(1675, 1048568, F16, 16) (dual of [1048568, 1048493, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(1675, 1048575, F16, 16) (dual of [1048575, 1048500, 17]-code), using
- 1 times truncation [i] based on linear OA(1676, 1048576, F16, 17) (dual of [1048576, 1048500, 18]-code), using
- an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 1 times truncation [i] based on linear OA(1676, 1048576, F16, 17) (dual of [1048576, 1048500, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(1675, 1048575, F16, 16) (dual of [1048575, 1048500, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(1675, 1048568, F16, 16) (dual of [1048568, 1048493, 17]-code), using
- net defined by OOA [i] based on linear OOA(1675, 131071, F16, 16, 16) (dual of [(131071, 16), 2097061, 17]-NRT-code), using
- digital (5, 13, 49)-net over F16, using
(72, 72+16, 262146)-Net in Base 16 — Constructive
(72, 88, 262146)-net in base 16, using
- 161 times duplication [i] based on (71, 87, 262146)-net in base 16, using
- net defined by OOA [i] based on OOA(1687, 262146, S16, 16, 16), using
- OA 8-folding and stacking [i] based on OA(1687, 2097168, S16, 16), using
- 1 times code embedding in larger space [i] based on OA(1686, 2097167, S16, 16), using
- discarding parts of the base [i] based on linear OA(12849, 2097167, F128, 16) (dual of [2097167, 2097118, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- linear OA(12846, 2097152, F128, 16) (dual of [2097152, 2097106, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(12834, 2097152, F128, 12) (dual of [2097152, 2097118, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(15) ⊂ Ce(11) [i] based on
- discarding parts of the base [i] based on linear OA(12849, 2097167, F128, 16) (dual of [2097167, 2097118, 17]-code), using
- 1 times code embedding in larger space [i] based on OA(1686, 2097167, S16, 16), using
- OA 8-folding and stacking [i] based on OA(1687, 2097168, S16, 16), using
- net defined by OOA [i] based on OOA(1687, 262146, S16, 16, 16), using
(72, 72+16, 4964169)-Net over F16 — Digital
Digital (72, 88, 4964169)-net over F16, using
(72, 72+16, large)-Net in Base 16 — Upper bound on s
There is no (72, 88, large)-net in base 16, because
- 14 times m-reduction [i] would yield (72, 74, large)-net in base 16, but