Best Known (22, 22+10, s)-Nets in Base 128
(22, 22+10, 419434)-Net over F128 — Constructive and digital
Digital (22, 32, 419434)-net over F128, using
- net defined by OOA [i] based on linear OOA(12832, 419434, F128, 10, 10) (dual of [(419434, 10), 4194308, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(12832, 2097170, F128, 10) (dual of [2097170, 2097138, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(12832, 2097171, F128, 10) (dual of [2097171, 2097139, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(4) [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(12813, 2097152, F128, 5) (dual of [2097152, 2097139, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(1284, 19, F128, 4) (dual of [19, 15, 5]-code or 19-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(9) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(12832, 2097171, F128, 10) (dual of [2097171, 2097139, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(12832, 2097170, F128, 10) (dual of [2097170, 2097138, 11]-code), using
(22, 22+10, 1677720)-Net in Base 128 — Constructive
(22, 32, 1677720)-net in base 128, using
- base change [i] based on digital (18, 28, 1677720)-net over F256, using
- net defined by OOA [i] based on linear OOA(25628, 1677720, F256, 10, 10) (dual of [(1677720, 10), 16777172, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(25628, 8388600, F256, 10) (dual of [8388600, 8388572, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(25628, large, F256, 10) (dual of [large, large−28, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(25628, large, F256, 10) (dual of [large, large−28, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(25628, 8388600, F256, 10) (dual of [8388600, 8388572, 11]-code), using
- net defined by OOA [i] based on linear OOA(25628, 1677720, F256, 10, 10) (dual of [(1677720, 10), 16777172, 11]-NRT-code), using
(22, 22+10, 2097171)-Net over F128 — Digital
Digital (22, 32, 2097171)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12832, 2097171, F128, 10) (dual of [2097171, 2097139, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(4) [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(12813, 2097152, F128, 5) (dual of [2097152, 2097139, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(1284, 19, F128, 4) (dual of [19, 15, 5]-code or 19-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(9) ⊂ Ce(4) [i] based on
(22, 22+10, 4194301)-Net in Base 128
(22, 32, 4194301)-net in base 128, using
- base change [i] based on digital (18, 28, 4194301)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25628, 4194301, F256, 2, 10) (dual of [(4194301, 2), 8388574, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25628, 8388602, F256, 10) (dual of [8388602, 8388574, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(25628, large, F256, 10) (dual of [large, large−28, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(25628, large, F256, 10) (dual of [large, large−28, 11]-code), using
- OOA 2-folding [i] based on linear OA(25628, 8388602, F256, 10) (dual of [8388602, 8388574, 11]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25628, 4194301, F256, 2, 10) (dual of [(4194301, 2), 8388574, 11]-NRT-code), using
(22, 22+10, large)-Net in Base 128 — Upper bound on s
There is no (22, 32, large)-net in base 128, because
- 8 times m-reduction [i] would yield (22, 24, large)-net in base 128, but