Best Known (40−16, 40, s)-Nets in Base 128
(40−16, 40, 2198)-Net over F128 — Constructive and digital
Digital (24, 40, 2198)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (1, 9, 150)-net over F128, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 1 and N(F) ≥ 150, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- digital (15, 31, 2048)-net over F128, using
- net defined by OOA [i] based on linear OOA(12831, 2048, F128, 16, 16) (dual of [(2048, 16), 32737, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(12831, 16384, F128, 16) (dual of [16384, 16353, 17]-code), using
- an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- OA 8-folding and stacking [i] based on linear OA(12831, 16384, F128, 16) (dual of [16384, 16353, 17]-code), using
- net defined by OOA [i] based on linear OOA(12831, 2048, F128, 16, 16) (dual of [(2048, 16), 32737, 17]-NRT-code), using
- digital (1, 9, 150)-net over F128, using
(40−16, 40, 8193)-Net in Base 128 — Constructive
(24, 40, 8193)-net in base 128, using
- base change [i] based on digital (19, 35, 8193)-net over F256, using
- 1 times m-reduction [i] based on digital (19, 36, 8193)-net over F256, using
- net defined by OOA [i] based on linear OOA(25636, 8193, F256, 17, 17) (dual of [(8193, 17), 139245, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(25636, 65545, F256, 17) (dual of [65545, 65509, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(25636, 65548, F256, 17) (dual of [65548, 65512, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- linear OA(25633, 65537, F256, 17) (dual of [65537, 65504, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(25625, 65537, F256, 13) (dual of [65537, 65512, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(2563, 11, F256, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,256) or 11-cap in PG(2,256)), using
- discarding factors / shortening the dual code based on linear OA(2563, 256, F256, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
- Reed–Solomon code RS(253,256) [i]
- discarding factors / shortening the dual code based on linear OA(2563, 256, F256, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25636, 65548, F256, 17) (dual of [65548, 65512, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(25636, 65545, F256, 17) (dual of [65545, 65509, 18]-code), using
- net defined by OOA [i] based on linear OOA(25636, 8193, F256, 17, 17) (dual of [(8193, 17), 139245, 18]-NRT-code), using
- 1 times m-reduction [i] based on digital (19, 36, 8193)-net over F256, using
(40−16, 40, 21054)-Net over F128 — Digital
Digital (24, 40, 21054)-net over F128, using
(40−16, 40, 28841)-Net in Base 128
(24, 40, 28841)-net in base 128, using
- base change [i] based on digital (19, 35, 28841)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25635, 28841, F256, 2, 16) (dual of [(28841, 2), 57647, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(25635, 32775, F256, 2, 16) (dual of [(32775, 2), 65515, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25635, 65550, F256, 16) (dual of [65550, 65515, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- linear OA(25631, 65536, F256, 16) (dual of [65536, 65505, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(25621, 65536, F256, 11) (dual of [65536, 65515, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(2564, 14, F256, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,256)), using
- discarding factors / shortening the dual code based on linear OA(2564, 256, F256, 4) (dual of [256, 252, 5]-code or 256-arc in PG(3,256)), using
- Reed–Solomon code RS(252,256) [i]
- discarding factors / shortening the dual code based on linear OA(2564, 256, F256, 4) (dual of [256, 252, 5]-code or 256-arc in PG(3,256)), using
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- OOA 2-folding [i] based on linear OA(25635, 65550, F256, 16) (dual of [65550, 65515, 17]-code), using
- discarding factors / shortening the dual code based on linear OOA(25635, 32775, F256, 2, 16) (dual of [(32775, 2), 65515, 17]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25635, 28841, F256, 2, 16) (dual of [(28841, 2), 57647, 17]-NRT-code), using
(40−16, 40, large)-Net in Base 128 — Upper bound on s
There is no (24, 40, large)-net in base 128, because
- 14 times m-reduction [i] would yield (24, 26, large)-net in base 128, but