Best Known (21, 31, s)-Nets in Base 128
(21, 31, 419433)-Net over F128 — Constructive and digital
Digital (21, 31, 419433)-net over F128, using
- net defined by OOA [i] based on linear OOA(12831, 419433, F128, 10, 10) (dual of [(419433, 10), 4194299, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(12831, 2097165, F128, 10) (dual of [2097165, 2097134, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(12831, 2097167, F128, 10) (dual of [2097167, 2097136, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(5) [i] based on
- 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(12816, 2097152, F128, 6) (dual of [2097152, 2097136, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(1283, 15, F128, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,128) or 15-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 Ce(9) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(12831, 2097167, F128, 10) (dual of [2097167, 2097136, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(12831, 2097165, F128, 10) (dual of [2097165, 2097134, 11]-code), using
(21, 31, 2097167)-Net over F128 — Digital
Digital (21, 31, 2097167)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12831, 2097167, F128, 10) (dual of [2097167, 2097136, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(5) [i] based on
- 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(12816, 2097152, F128, 6) (dual of [2097152, 2097136, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(1283, 15, F128, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,128) or 15-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 Ce(9) ⊂ Ce(5) [i] based on
(21, 31, large)-Net in Base 128 — Upper bound on s
There is no (21, 31, large)-net in base 128, because
- 8 times m-reduction [i] would yield (21, 23, large)-net in base 128, but