Best Known (40−18, 40, s)-Nets in Base 128
(40−18, 40, 1822)-Net over F128 — Constructive and digital
Digital (22, 40, 1822)-net over F128, using
- 1281 times duplication [i] based on digital (21, 39, 1822)-net over F128, using
- net defined by OOA [i] based on linear OOA(12839, 1822, F128, 18, 18) (dual of [(1822, 18), 32757, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(12839, 16398, F128, 18) (dual of [16398, 16359, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- linear OA(12835, 16384, F128, 18) (dual of [16384, 16349, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(12825, 16384, F128, 13) (dual of [16384, 16359, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(17) ⊂ Ce(12) [i] based on
- OA 9-folding and stacking [i] based on linear OA(12839, 16398, F128, 18) (dual of [16398, 16359, 19]-code), using
- net defined by OOA [i] based on linear OOA(12839, 1822, F128, 18, 18) (dual of [(1822, 18), 32757, 19]-NRT-code), using
(40−18, 40, 7282)-Net in Base 128 — Constructive
(22, 40, 7282)-net in base 128, using
- base change [i] based on digital (17, 35, 7282)-net over F256, using
- net defined by OOA [i] based on linear OOA(25635, 7282, F256, 18, 18) (dual of [(7282, 18), 131041, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(25635, 65538, F256, 18) (dual of [65538, 65503, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(25635, 65536, F256, 18) (dual of [65536, 65501, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(25633, 65536, F256, 17) (dual of [65536, 65503, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(17) ⊂ Ce(16) [i] based on
- OA 9-folding and stacking [i] based on linear OA(25635, 65538, F256, 18) (dual of [65538, 65503, 19]-code), using
- net defined by OOA [i] based on linear OOA(25635, 7282, F256, 18, 18) (dual of [(7282, 18), 131041, 19]-NRT-code), using
(40−18, 40, 8200)-Net over F128 — Digital
Digital (22, 40, 8200)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12840, 8200, F128, 2, 18) (dual of [(8200, 2), 16360, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12840, 16400, F128, 18) (dual of [16400, 16360, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(12840, 16401, F128, 18) (dual of [16401, 16361, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(11) [i] based on
- linear OA(12835, 16384, F128, 18) (dual of [16384, 16349, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- 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(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 Ce(17) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(12840, 16401, F128, 18) (dual of [16401, 16361, 19]-code), using
- OOA 2-folding [i] based on linear OA(12840, 16400, F128, 18) (dual of [16400, 16360, 19]-code), using
(40−18, 40, 13107)-Net in Base 128
(22, 40, 13107)-net in base 128, using
- base change [i] based on digital (17, 35, 13107)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25635, 13107, F256, 5, 18) (dual of [(13107, 5), 65500, 19]-NRT-code), using
- OOA 5-folding [i] based on linear OA(25635, 65535, F256, 18) (dual of [65535, 65500, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(25635, 65536, F256, 18) (dual of [65536, 65501, 19]-code), using
- an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(25635, 65536, F256, 18) (dual of [65536, 65501, 19]-code), using
- OOA 5-folding [i] based on linear OA(25635, 65535, F256, 18) (dual of [65535, 65500, 19]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25635, 13107, F256, 5, 18) (dual of [(13107, 5), 65500, 19]-NRT-code), using
(40−18, 40, large)-Net in Base 128 — Upper bound on s
There is no (22, 40, large)-net in base 128, because
- 16 times m-reduction [i] would yield (22, 24, large)-net in base 128, but