Best Known (15, 15+12, s)-Nets in Base 128
(15, 15+12, 2733)-Net over F128 — Constructive and digital
Digital (15, 27, 2733)-net over F128, using
- net defined by OOA [i] based on linear OOA(12827, 2733, F128, 12, 12) (dual of [(2733, 12), 32769, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(12827, 16398, F128, 12) (dual of [16398, 16371, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(6) [i] based on
- linear OA(12823, 16384, F128, 12) (dual of [16384, 16361, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(12813, 16384, F128, 7) (dual of [16384, 16371, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(11) ⊂ Ce(6) [i] based on
- OA 6-folding and stacking [i] based on linear OA(12827, 16398, F128, 12) (dual of [16398, 16371, 13]-code), using
(15, 15+12, 10734)-Net over F128 — Digital
Digital (15, 27, 10734)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12827, 10734, F128, 12) (dual of [10734, 10707, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(12827, 16398, F128, 12) (dual of [16398, 16371, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(6) [i] based on
- linear OA(12823, 16384, F128, 12) (dual of [16384, 16361, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(12813, 16384, F128, 7) (dual of [16384, 16371, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(11) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(12827, 16398, F128, 12) (dual of [16398, 16371, 13]-code), using
(15, 15+12, 10923)-Net in Base 128 — Constructive
(15, 27, 10923)-net in base 128, using
- net defined by OOA [i] based on OOA(12827, 10923, S128, 12, 12), using
- OA 6-folding and stacking [i] based on OA(12827, 65538, S128, 12), using
- discarding parts of the base [i] based on linear OA(25623, 65538, F256, 12) (dual of [65538, 65515, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(25623, 65536, F256, 12) (dual of [65536, 65513, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- 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(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(11) ⊂ Ce(10) [i] based on
- discarding parts of the base [i] based on linear OA(25623, 65538, F256, 12) (dual of [65538, 65515, 13]-code), using
- OA 6-folding and stacking [i] based on OA(12827, 65538, S128, 12), using
(15, 15+12, large)-Net in Base 128 — Upper bound on s
There is no (15, 27, large)-net in base 128, because
- 10 times m-reduction [i] would yield (15, 17, large)-net in base 128, but