Best Known (23−10, 23, s)-Nets in Base 128
(23−10, 23, 3279)-Net over F128 — Constructive and digital
Digital (13, 23, 3279)-net over F128, using
- 1 times m-reduction [i] based on digital (13, 24, 3279)-net over F128, using
- net defined by OOA [i] based on linear OOA(12824, 3279, F128, 11, 11) (dual of [(3279, 11), 36045, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(12824, 16396, F128, 11) (dual of [16396, 16372, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,3]) [i] based on
- 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(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(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([0,5]) ⊂ C([0,3]) [i] based on
- OOA 5-folding and stacking with additional row [i] based on linear OA(12824, 16396, F128, 11) (dual of [16396, 16372, 12]-code), using
- net defined by OOA [i] based on linear OOA(12824, 3279, F128, 11, 11) (dual of [(3279, 11), 36045, 12]-NRT-code), using
(23−10, 23, 13108)-Net in Base 128 — Constructive
(13, 23, 13108)-net in base 128, using
- net defined by OOA [i] based on OOA(12823, 13108, S128, 10, 10), using
- OA 5-folding and stacking [i] based on OA(12823, 65540, S128, 10), using
- discarding factors based on OA(12823, 65541, S128, 10), using
- discarding parts of the base [i] based on linear OA(25620, 65541, F256, 10) (dual of [65541, 65521, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(25619, 65536, F256, 10) (dual of [65536, 65517, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(25615, 65536, F256, 8) (dual of [65536, 65521, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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 Ce(9) ⊂ Ce(7) [i] based on
- discarding parts of the base [i] based on linear OA(25620, 65541, F256, 10) (dual of [65541, 65521, 11]-code), using
- discarding factors based on OA(12823, 65541, S128, 10), using
- OA 5-folding and stacking [i] based on OA(12823, 65540, S128, 10), using
(23−10, 23, 16398)-Net over F128 — Digital
Digital (13, 23, 16398)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12823, 16398, F128, 10) (dual of [16398, 16375, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(4) [i] based on
- linear OA(12819, 16384, F128, 10) (dual of [16384, 16365, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(1289, 16384, F128, 5) (dual of [16384, 16375, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(9) ⊂ Ce(4) [i] based on
(23−10, 23, 20651)-Net in Base 128
(13, 23, 20651)-net in base 128, using
- 1 times m-reduction [i] based on (13, 24, 20651)-net in base 128, using
- base change [i] based on digital (10, 21, 20651)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25621, 20651, F256, 3, 11) (dual of [(20651, 3), 61932, 12]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(25621, 21846, F256, 3, 11) (dual of [(21846, 3), 65517, 12]-NRT-code), using
- OOA 3-folding [i] based on linear OA(25621, 65538, F256, 11) (dual of [65538, 65517, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(25621, 65536, F256, 11) (dual of [65536, 65515, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(25619, 65536, F256, 10) (dual of [65536, 65517, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(10) ⊂ Ce(9) [i] based on
- OOA 3-folding [i] based on linear OA(25621, 65538, F256, 11) (dual of [65538, 65517, 12]-code), using
- discarding factors / shortening the dual code based on linear OOA(25621, 21846, F256, 3, 11) (dual of [(21846, 3), 65517, 12]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25621, 20651, F256, 3, 11) (dual of [(20651, 3), 61932, 12]-NRT-code), using
- base change [i] based on digital (10, 21, 20651)-net over F256, using
(23−10, 23, large)-Net in Base 128 — Upper bound on s
There is no (13, 23, large)-net in base 128, because
- 8 times m-reduction [i] would yield (13, 15, large)-net in base 128, but