Best Known (13, 22, s)-Nets in Base 128
(13, 22, 4246)-Net over F128 — Constructive and digital
Digital (13, 22, 4246)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (1, 5, 150)-net over F128, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 1 and N(F) ≥ 150, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- digital (8, 17, 4096)-net over F128, using
- net defined by OOA [i] based on linear OOA(12817, 4096, F128, 9, 9) (dual of [(4096, 9), 36847, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(12817, 16385, F128, 9) (dual of [16385, 16368, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- OOA 4-folding and stacking with additional row [i] based on linear OA(12817, 16385, F128, 9) (dual of [16385, 16368, 10]-code), using
- net defined by OOA [i] based on linear OOA(12817, 4096, F128, 9, 9) (dual of [(4096, 9), 36847, 10]-NRT-code), using
- digital (1, 5, 150)-net over F128, using
(13, 22, 16385)-Net in Base 128 — Constructive
(13, 22, 16385)-net in base 128, using
- 1281 times duplication [i] based on (12, 21, 16385)-net in base 128, using
- net defined by OOA [i] based on OOA(12821, 16385, S128, 9, 9), using
- OOA 4-folding and stacking with additional row [i] based on OA(12821, 65541, S128, 9), using
- discarding factors based on OA(12821, 65542, S128, 9), using
- discarding parts of the base [i] based on linear OA(25618, 65542, F256, 9) (dual of [65542, 65524, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(25617, 65537, F256, 9) (dual of [65537, 65520, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(25613, 65537, F256, 7) (dual of [65537, 65524, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(2561, 5, F256, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, s, F256, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- discarding parts of the base [i] based on linear OA(25618, 65542, F256, 9) (dual of [65542, 65524, 10]-code), using
- discarding factors based on OA(12821, 65542, S128, 9), using
- OOA 4-folding and stacking with additional row [i] based on OA(12821, 65541, S128, 9), using
- net defined by OOA [i] based on OOA(12821, 16385, S128, 9, 9), using
(13, 22, 18484)-Net over F128 — Digital
Digital (13, 22, 18484)-net over F128, using
(13, 22, 32771)-Net in Base 128
(13, 22, 32771)-net in base 128, using
- net defined by OOA [i] based on OOA(12822, 32771, S128, 12, 9), using
- OOA stacking with additional row [i] based on OOA(12822, 32772, S128, 4, 9), using
- discarding parts of the base [i] based on linear OOA(25619, 32772, F256, 4, 9) (dual of [(32772, 4), 131069, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25619, 32772, F256, 2, 9) (dual of [(32772, 2), 65525, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25619, 65544, F256, 9) (dual of [65544, 65525, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(25617, 65536, F256, 9) (dual of [65536, 65519, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(25611, 65536, F256, 6) (dual of [65536, 65525, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(2562, 8, F256, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,256)), using
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- Reed–Solomon code RS(254,256) [i]
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- OOA 2-folding [i] based on linear OA(25619, 65544, F256, 9) (dual of [65544, 65525, 10]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25619, 32772, F256, 2, 9) (dual of [(32772, 2), 65525, 10]-NRT-code), using
- discarding parts of the base [i] based on linear OOA(25619, 32772, F256, 4, 9) (dual of [(32772, 4), 131069, 10]-NRT-code), using
- OOA stacking with additional row [i] based on OOA(12822, 32772, S128, 4, 9), using
(13, 22, large)-Net in Base 128 — Upper bound on s
There is no (13, 22, large)-net in base 128, because
- 7 times m-reduction [i] would yield (13, 15, large)-net in base 128, but