Best Known (23, 39, s)-Nets in Base 128
(23, 39, 2177)-Net over F128 — Constructive and digital
Digital (23, 39, 2177)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (0, 8, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-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 (0, 8, 129)-net over F128, using
(23, 39, 8193)-Net in Base 128 — Constructive
(23, 39, 8193)-net in base 128, using
- 1281 times duplication [i] based on (22, 38, 8193)-net in base 128, using
- net defined by OOA [i] based on OOA(12838, 8193, S128, 16, 16), using
- OA 8-folding and stacking [i] based on OA(12838, 65544, S128, 16), using
- discarding parts of the base [i] based on linear OA(25633, 65544, F256, 16) (dual of [65544, 65511, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [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(25625, 65536, F256, 13) (dual of [65536, 65511, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(2562, 8, F256, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,256)), using
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- Reed–Solomon code RS(254,256) [i]
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- discarding parts of the base [i] based on linear OA(25633, 65544, F256, 16) (dual of [65544, 65511, 17]-code), using
- OA 8-folding and stacking [i] based on OA(12838, 65544, S128, 16), using
- net defined by OOA [i] based on OOA(12838, 8193, S128, 16, 16), using
(23, 39, 16515)-Net over F128 — Digital
Digital (23, 39, 16515)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12839, 16515, F128, 16) (dual of [16515, 16476, 17]-code), using
- (u, u+v)-construction [i] based on
- linear OA(1288, 129, F128, 8) (dual of [129, 121, 9]-code or 129-arc in PG(7,128)), using
- extended Reed–Solomon code RSe(121,128) [i]
- linear OA(12831, 16386, F128, 16) (dual of [16386, 16355, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(14) [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]
- linear OA(12829, 16384, F128, 15) (dual of [16384, 16355, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(1280, 2, F128, 0) (dual of [2, 2, 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(15) ⊂ Ce(14) [i] based on
- linear OA(1288, 129, F128, 8) (dual of [129, 121, 9]-code or 129-arc in PG(7,128)), using
- (u, u+v)-construction [i] based on
(23, 39, 18824)-Net in Base 128
(23, 39, 18824)-net in base 128, using
- 1 times m-reduction [i] based on (23, 40, 18824)-net in base 128, using
- base change [i] based on digital (18, 35, 18824)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25635, 18824, F256, 3, 17) (dual of [(18824, 3), 56437, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(25635, 21848, F256, 3, 17) (dual of [(21848, 3), 65509, 18]-NRT-code), using
- OOA 3-folding [i] based on linear OA(25635, 65544, F256, 17) (dual of [65544, 65509, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- linear OA(25633, 65536, F256, 17) (dual of [65536, 65503, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(25627, 65536, F256, 14) (dual of [65536, 65509, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(2562, 8, F256, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,256)), using
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- Reed–Solomon code RS(254,256) [i]
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- OOA 3-folding [i] based on linear OA(25635, 65544, F256, 17) (dual of [65544, 65509, 18]-code), using
- discarding factors / shortening the dual code based on linear OOA(25635, 21848, F256, 3, 17) (dual of [(21848, 3), 65509, 18]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25635, 18824, F256, 3, 17) (dual of [(18824, 3), 56437, 18]-NRT-code), using
- base change [i] based on digital (18, 35, 18824)-net over F256, using
(23, 39, large)-Net in Base 128 — Upper bound on s
There is no (23, 39, large)-net in base 128, because
- 14 times m-reduction [i] would yield (23, 25, large)-net in base 128, but