Best Known (27, 27+21, s)-Nets in Base 128
(27, 27+21, 1640)-Net over F128 — Constructive and digital
Digital (27, 48, 1640)-net over F128, using
- 1282 times duplication [i] based on digital (25, 46, 1640)-net over F128, using
- net defined by OOA [i] based on linear OOA(12846, 1640, F128, 21, 21) (dual of [(1640, 21), 34394, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(12846, 16401, F128, 21) (dual of [16401, 16355, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(12846, 16402, F128, 21) (dual of [16402, 16356, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,7]) [i] based on
- linear OA(12841, 16385, F128, 21) (dual of [16385, 16344, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(12829, 16385, F128, 15) (dual of [16385, 16356, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(1285, 17, F128, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,128)), using
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- Reed–Solomon code RS(123,128) [i]
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- construction X applied to C([0,10]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12846, 16402, F128, 21) (dual of [16402, 16356, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(12846, 16401, F128, 21) (dual of [16401, 16355, 22]-code), using
- net defined by OOA [i] based on linear OOA(12846, 1640, F128, 21, 21) (dual of [(1640, 21), 34394, 22]-NRT-code), using
(27, 27+21, 6554)-Net in Base 128 — Constructive
(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
(27, 27+21, 10177)-Net over F128 — Digital
Digital (27, 48, 10177)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12848, 10177, F128, 21) (dual of [10177, 10129, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(12848, 16408, F128, 21) (dual of [16408, 16360, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,6]) [i] based on
- linear OA(12841, 16385, F128, 21) (dual of [16385, 16344, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(12825, 16385, F128, 13) (dual of [16385, 16360, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(1287, 23, F128, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,128)), using
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- Reed–Solomon code RS(121,128) [i]
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- construction X applied to C([0,10]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12848, 16408, F128, 21) (dual of [16408, 16360, 22]-code), using
(27, 27+21, 13975)-Net in Base 128
(27, 48, 13975)-net in base 128, using
- base change [i] based on digital (21, 42, 13975)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25642, 13975, F256, 4, 21) (dual of [(13975, 4), 55858, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(25642, 16385, F256, 4, 21) (dual of [(16385, 4), 65498, 22]-NRT-code), using
- OOA 4-folding [i] based on linear OA(25642, 65540, F256, 21) (dual of [65540, 65498, 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 4-folding [i] based on linear OA(25642, 65540, F256, 21) (dual of [65540, 65498, 22]-code), using
- discarding factors / shortening the dual code based on linear OOA(25642, 16385, F256, 4, 21) (dual of [(16385, 4), 65498, 22]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25642, 13975, F256, 4, 21) (dual of [(13975, 4), 55858, 22]-NRT-code), using
(27, 27+21, large)-Net in Base 128 — Upper bound on s
There is no (27, 48, large)-net in base 128, because
- 19 times m-reduction [i] would yield (27, 29, large)-net in base 128, but