Best Known (60−25, 60, s)-Nets in Base 128
(60−25, 60, 1368)-Net over F128 — Constructive and digital
Digital (35, 60, 1368)-net over F128, using
- net defined by OOA [i] based on linear OOA(12860, 1368, F128, 25, 25) (dual of [(1368, 25), 34140, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(12860, 16417, F128, 25) (dual of [16417, 16357, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(12860, 16420, F128, 25) (dual of [16420, 16360, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,6]) [i] based on
- linear OA(12849, 16385, F128, 25) (dual of [16385, 16336, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (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(12811, 35, F128, 11) (dual of [35, 24, 12]-code or 35-arc in PG(10,128)), using
- discarding factors / shortening the dual code based on linear OA(12811, 128, F128, 11) (dual of [128, 117, 12]-code or 128-arc in PG(10,128)), using
- Reed–Solomon code RS(117,128) [i]
- discarding factors / shortening the dual code based on linear OA(12811, 128, F128, 11) (dual of [128, 117, 12]-code or 128-arc in PG(10,128)), using
- construction X applied to C([0,12]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12860, 16420, F128, 25) (dual of [16420, 16360, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(12860, 16417, F128, 25) (dual of [16417, 16357, 26]-code), using
(60−25, 60, 5462)-Net in Base 128 — Constructive
(35, 60, 5462)-net in base 128, using
- net defined by OOA [i] based on OOA(12860, 5462, S128, 25, 25), using
- OOA 12-folding and stacking with additional row [i] based on OA(12860, 65545, S128, 25), using
- discarding factors based on OA(12860, 65548, S128, 25), using
- discarding parts of the base [i] based on linear OA(25652, 65548, F256, 25) (dual of [65548, 65496, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(25649, 65537, F256, 25) (dual of [65537, 65488, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 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(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 C([0,12]) ⊂ C([0,10]) [i] based on
- discarding parts of the base [i] based on linear OA(25652, 65548, F256, 25) (dual of [65548, 65496, 26]-code), using
- discarding factors based on OA(12860, 65548, S128, 25), using
- OOA 12-folding and stacking with additional row [i] based on OA(12860, 65545, S128, 25), using
(60−25, 60, 16420)-Net over F128 — Digital
Digital (35, 60, 16420)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12860, 16420, F128, 25) (dual of [16420, 16360, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,6]) [i] based on
- linear OA(12849, 16385, F128, 25) (dual of [16385, 16336, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (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(12811, 35, F128, 11) (dual of [35, 24, 12]-code or 35-arc in PG(10,128)), using
- discarding factors / shortening the dual code based on linear OA(12811, 128, F128, 11) (dual of [128, 117, 12]-code or 128-arc in PG(10,128)), using
- Reed–Solomon code RS(117,128) [i]
- discarding factors / shortening the dual code based on linear OA(12811, 128, F128, 11) (dual of [128, 117, 12]-code or 128-arc in PG(10,128)), using
- construction X applied to C([0,12]) ⊂ C([0,6]) [i] based on
(60−25, 60, large)-Net in Base 128 — Upper bound on s
There is no (35, 60, large)-net in base 128, because
- 23 times m-reduction [i] would yield (35, 37, large)-net in base 128, but