Best Known (77−23, 77, s)-Nets in Base 128
(77−23, 77, 190654)-Net over F128 — Constructive and digital
Digital (54, 77, 190654)-net over F128, using
- net defined by OOA [i] based on linear OOA(12877, 190654, F128, 23, 23) (dual of [(190654, 23), 4384965, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(12877, 2097195, F128, 23) (dual of [2097195, 2097118, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(11) [i] based on
- linear OA(12867, 2097152, F128, 23) (dual of [2097152, 2097085, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(12834, 2097152, F128, 12) (dual of [2097152, 2097118, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(12810, 43, F128, 10) (dual of [43, 33, 11]-code or 43-arc in PG(9,128)), using
- discarding factors / shortening the dual code based on linear OA(12810, 128, F128, 10) (dual of [128, 118, 11]-code or 128-arc in PG(9,128)), using
- Reed–Solomon code RS(118,128) [i]
- discarding factors / shortening the dual code based on linear OA(12810, 128, F128, 10) (dual of [128, 118, 11]-code or 128-arc in PG(9,128)), using
- construction X applied to Ce(22) ⊂ Ce(11) [i] based on
- OOA 11-folding and stacking with additional row [i] based on linear OA(12877, 2097195, F128, 23) (dual of [2097195, 2097118, 24]-code), using
(77−23, 77, 762600)-Net in Base 128 — Constructive
(54, 77, 762600)-net in base 128, using
- net defined by OOA [i] based on OOA(12877, 762600, S128, 23, 23), using
- OOA 11-folding and stacking with additional row [i] based on OA(12877, 8388601, S128, 23), using
- discarding factors based on OA(12877, large, S128, 23), using
- discarding parts of the base [i] based on linear OA(25667, large, F256, 23) (dual of [large, large−67, 24]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,22], and designed minimum distance d ≥ |I|+1 = 24 [i]
- discarding parts of the base [i] based on linear OA(25667, large, F256, 23) (dual of [large, large−67, 24]-code), using
- discarding factors based on OA(12877, large, S128, 23), using
- OOA 11-folding and stacking with additional row [i] based on OA(12877, 8388601, S128, 23), using
(77−23, 77, 2097195)-Net over F128 — Digital
Digital (54, 77, 2097195)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12877, 2097195, F128, 23) (dual of [2097195, 2097118, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(11) [i] based on
- linear OA(12867, 2097152, F128, 23) (dual of [2097152, 2097085, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(12834, 2097152, F128, 12) (dual of [2097152, 2097118, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(12810, 43, F128, 10) (dual of [43, 33, 11]-code or 43-arc in PG(9,128)), using
- discarding factors / shortening the dual code based on linear OA(12810, 128, F128, 10) (dual of [128, 118, 11]-code or 128-arc in PG(9,128)), using
- Reed–Solomon code RS(118,128) [i]
- discarding factors / shortening the dual code based on linear OA(12810, 128, F128, 10) (dual of [128, 118, 11]-code or 128-arc in PG(9,128)), using
- construction X applied to Ce(22) ⊂ Ce(11) [i] based on
(77−23, 77, large)-Net in Base 128 — Upper bound on s
There is no (54, 77, large)-net in base 128, because
- 21 times m-reduction [i] would yield (54, 56, large)-net in base 128, but