Best Known (60−18, 60, s)-Nets in Base 128
(60−18, 60, 233020)-Net over F128 — Constructive and digital
Digital (42, 60, 233020)-net over F128, using
- 1281 times duplication [i] based on digital (41, 59, 233020)-net over F128, using
- net defined by OOA [i] based on linear OOA(12859, 233020, F128, 18, 18) (dual of [(233020, 18), 4194301, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(12859, 2097180, F128, 18) (dual of [2097180, 2097121, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(12859, 2097183, F128, 18) (dual of [2097183, 2097124, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(9) [i] based on
- linear OA(12852, 2097152, F128, 18) (dual of [2097152, 2097100, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(12828, 2097152, F128, 10) (dual of [2097152, 2097124, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(1287, 31, F128, 7) (dual of [31, 24, 8]-code or 31-arc in PG(6,128)), using
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- Reed–Solomon code RS(121,128) [i]
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- construction X applied to Ce(17) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(12859, 2097183, F128, 18) (dual of [2097183, 2097124, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(12859, 2097180, F128, 18) (dual of [2097180, 2097121, 19]-code), using
- net defined by OOA [i] based on linear OOA(12859, 233020, F128, 18, 18) (dual of [(233020, 18), 4194301, 19]-NRT-code), using
(60−18, 60, 932067)-Net in Base 128 — Constructive
(42, 60, 932067)-net in base 128, using
- net defined by OOA [i] based on OOA(12860, 932067, S128, 18, 18), using
- OA 9-folding and stacking [i] based on OA(12860, large, S128, 18), using
- discarding parts of the base [i] based on linear OA(25652, large, F256, 18) (dual of [large, large−52, 19]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- discarding parts of the base [i] based on linear OA(25652, large, F256, 18) (dual of [large, large−52, 19]-code), using
- OA 9-folding and stacking [i] based on OA(12860, large, S128, 18), using
(60−18, 60, 2097187)-Net over F128 — Digital
Digital (42, 60, 2097187)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12860, 2097187, F128, 18) (dual of [2097187, 2097127, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(8) [i] based on
- linear OA(12852, 2097152, F128, 18) (dual of [2097152, 2097100, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(12825, 2097152, F128, 9) (dual of [2097152, 2097127, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(1288, 35, F128, 8) (dual of [35, 27, 9]-code or 35-arc in PG(7,128)), using
- discarding factors / shortening the dual code based on linear OA(1288, 128, F128, 8) (dual of [128, 120, 9]-code or 128-arc in PG(7,128)), using
- Reed–Solomon code RS(120,128) [i]
- discarding factors / shortening the dual code based on linear OA(1288, 128, F128, 8) (dual of [128, 120, 9]-code or 128-arc in PG(7,128)), using
- construction X applied to Ce(17) ⊂ Ce(8) [i] based on
(60−18, 60, large)-Net in Base 128 — Upper bound on s
There is no (42, 60, large)-net in base 128, because
- 16 times m-reduction [i] would yield (42, 44, large)-net in base 128, but