Best Known (17, 31, s)-Nets in Base 128
(17, 31, 2342)-Net over F128 — Constructive and digital
Digital (17, 31, 2342)-net over F128, using
- 1 times m-reduction [i] based on digital (17, 32, 2342)-net over F128, using
- net defined by OOA [i] based on linear OOA(12832, 2342, F128, 15, 15) (dual of [(2342, 15), 35098, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12832, 16395, F128, 15) (dual of [16395, 16363, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(12832, 16396, F128, 15) (dual of [16396, 16364, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- 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(12821, 16385, F128, 11) (dual of [16385, 16364, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(1283, 11, F128, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,128) or 11-cap in PG(2,128)), using
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- Reed–Solomon code RS(125,128) [i]
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12832, 16396, F128, 15) (dual of [16396, 16364, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12832, 16395, F128, 15) (dual of [16395, 16363, 16]-code), using
- net defined by OOA [i] based on linear OOA(12832, 2342, F128, 15, 15) (dual of [(2342, 15), 35098, 16]-NRT-code), using
(17, 31, 8199)-Net over F128 — Digital
Digital (17, 31, 8199)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12831, 8199, F128, 2, 14) (dual of [(8199, 2), 16367, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12831, 16398, F128, 14) (dual of [16398, 16367, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(8) [i] based on
- linear OA(12827, 16384, F128, 14) (dual of [16384, 16357, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(12817, 16384, F128, 9) (dual of [16384, 16367, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(1284, 14, F128, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,128)), using
- discarding factors / shortening the dual code based on linear OA(1284, 128, F128, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
- Reed–Solomon code RS(124,128) [i]
- discarding factors / shortening the dual code based on linear OA(1284, 128, F128, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
- construction X applied to Ce(13) ⊂ Ce(8) [i] based on
- OOA 2-folding [i] based on linear OA(12831, 16398, F128, 14) (dual of [16398, 16367, 15]-code), using
(17, 31, 9362)-Net in Base 128 — Constructive
(17, 31, 9362)-net in base 128, using
- net defined by OOA [i] based on OOA(12831, 9362, S128, 14, 14), using
- OA 7-folding and stacking [i] based on OA(12831, 65534, S128, 14), using
- discarding factors based on OA(12831, 65538, S128, 14), using
- discarding parts of the base [i] based on linear OA(25627, 65538, F256, 14) (dual of [65538, 65511, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- 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(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(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- discarding parts of the base [i] based on linear OA(25627, 65538, F256, 14) (dual of [65538, 65511, 15]-code), using
- discarding factors based on OA(12831, 65538, S128, 14), using
- OA 7-folding and stacking [i] based on OA(12831, 65534, S128, 14), using
(17, 31, large)-Net in Base 128 — Upper bound on s
There is no (17, 31, large)-net in base 128, because
- 12 times m-reduction [i] would yield (17, 19, large)-net in base 128, but