Best Known (29, 29+20, s)-Nets in Base 128
(29, 29+20, 1767)-Net over F128 — Constructive and digital
Digital (29, 49, 1767)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (0, 10, 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 (19, 39, 1638)-net over F128, using
- net defined by OOA [i] based on linear OOA(12839, 1638, F128, 20, 20) (dual of [(1638, 20), 32721, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(12839, 16380, F128, 20) (dual of [16380, 16341, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(12839, 16384, F128, 20) (dual of [16384, 16345, 21]-code), using
- an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(12839, 16384, F128, 20) (dual of [16384, 16345, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(12839, 16380, F128, 20) (dual of [16380, 16341, 21]-code), using
- net defined by OOA [i] based on linear OOA(12839, 1638, F128, 20, 20) (dual of [(1638, 20), 32721, 21]-NRT-code), using
- digital (0, 10, 129)-net over F128, using
(29, 29+20, 6554)-Net in Base 128 — Constructive
(29, 49, 6554)-net in base 128, using
- 1281 times duplication [i] based on (28, 48, 6554)-net in base 128, using
- t-expansion [i] based on (27, 48, 6554)-net in base 128, using
- base change [i] based on digital (21, 42, 6554)-net over F256, using
- net defined by OOA [i] based on linear OOA(25642, 6554, F256, 21, 21) (dual of [(6554, 21), 137592, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(25642, 65541, F256, 21) (dual of [65541, 65499, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(25642, 65542, F256, 21) (dual of [65542, 65500, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(25641, 65537, F256, 21) (dual of [65537, 65496, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(25637, 65537, F256, 19) (dual of [65537, 65500, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(2561, 5, F256, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, s, F256, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25642, 65542, F256, 21) (dual of [65542, 65500, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(25642, 65541, F256, 21) (dual of [65541, 65499, 22]-code), using
- net defined by OOA [i] based on linear OOA(25642, 6554, F256, 21, 21) (dual of [(6554, 21), 137592, 22]-NRT-code), using
- base change [i] based on digital (21, 42, 6554)-net over F256, using
- t-expansion [i] based on (27, 48, 6554)-net in base 128, using
(29, 29+20, 16984)-Net over F128 — Digital
Digital (29, 49, 16984)-net over F128, using
(29, 29+20, 19766)-Net in Base 128
(29, 49, 19766)-net in base 128, using
- 1281 times duplication [i] based on (28, 48, 19766)-net in base 128, using
- base change [i] based on digital (22, 42, 19766)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25642, 19766, F256, 3, 20) (dual of [(19766, 3), 59256, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(25642, 21849, F256, 3, 20) (dual of [(21849, 3), 65505, 21]-NRT-code), using
- OOA 3-folding [i] based on linear OA(25642, 65547, F256, 20) (dual of [65547, 65505, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(15) [i] based on
- linear OA(25639, 65536, F256, 20) (dual of [65536, 65497, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- 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(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 Ce(19) ⊂ Ce(15) [i] based on
- OOA 3-folding [i] based on linear OA(25642, 65547, F256, 20) (dual of [65547, 65505, 21]-code), using
- discarding factors / shortening the dual code based on linear OOA(25642, 21849, F256, 3, 20) (dual of [(21849, 3), 65505, 21]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25642, 19766, F256, 3, 20) (dual of [(19766, 3), 59256, 21]-NRT-code), using
- base change [i] based on digital (22, 42, 19766)-net over F256, using
(29, 29+20, large)-Net in Base 128 — Upper bound on s
There is no (29, 49, large)-net in base 128, because
- 18 times m-reduction [i] would yield (29, 31, large)-net in base 128, but