Best Known (30−13, 30, s)-Nets in Base 128
(30−13, 30, 2733)-Net over F128 — Constructive and digital
Digital (17, 30, 2733)-net over F128, using
- net defined by OOA [i] based on linear OOA(12830, 2733, F128, 13, 13) (dual of [(2733, 13), 35499, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(12830, 16399, F128, 13) (dual of [16399, 16369, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(12830, 16402, F128, 13) (dual of [16402, 16372, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,3]) [i] based on
- 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(12813, 16385, F128, 7) (dual of [16385, 16372, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (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([0,6]) ⊂ C([0,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12830, 16402, F128, 13) (dual of [16402, 16372, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(12830, 16399, F128, 13) (dual of [16399, 16369, 14]-code), using
(30−13, 30, 10923)-Net in Base 128 — Constructive
(17, 30, 10923)-net in base 128, using
- net defined by OOA [i] based on OOA(12830, 10923, S128, 13, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(12830, 65539, S128, 13), using
- discarding factors based on OA(12830, 65542, S128, 13), using
- discarding parts of the base [i] based on linear OA(25626, 65542, F256, 13) (dual of [65542, 65516, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- 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(25621, 65537, F256, 11) (dual of [65537, 65516, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (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([0,6]) ⊂ C([0,5]) [i] based on
- discarding parts of the base [i] based on linear OA(25626, 65542, F256, 13) (dual of [65542, 65516, 14]-code), using
- discarding factors based on OA(12830, 65542, S128, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(12830, 65539, S128, 13), using
(30−13, 30, 13882)-Net over F128 — Digital
Digital (17, 30, 13882)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12830, 13882, F128, 13) (dual of [13882, 13852, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(12830, 16402, F128, 13) (dual of [16402, 16372, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,3]) [i] based on
- 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(12813, 16385, F128, 7) (dual of [16385, 16372, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (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([0,6]) ⊂ C([0,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12830, 16402, F128, 13) (dual of [16402, 16372, 14]-code), using
(30−13, 30, large)-Net in Base 128 — Upper bound on s
There is no (17, 30, large)-net in base 128, because
- 11 times m-reduction [i] would yield (17, 19, large)-net in base 128, but