Best Known (11, 11+8, s)-Nets in Base 128
(11, 11+8, 4225)-Net over F128 — Constructive and digital
Digital (11, 19, 4225)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- digital (7, 15, 4096)-net over F128, using
- net defined by OOA [i] based on linear OOA(12815, 4096, F128, 8, 8) (dual of [(4096, 8), 32753, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(12815, 16384, F128, 8) (dual of [16384, 16369, 9]-code), using
- an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- OA 4-folding and stacking [i] based on linear OA(12815, 16384, F128, 8) (dual of [16384, 16369, 9]-code), using
- net defined by OOA [i] based on linear OOA(12815, 4096, F128, 8, 8) (dual of [(4096, 8), 32753, 9]-NRT-code), using
- digital (0, 4, 129)-net over F128, using
(11, 11+8, 16385)-Net in Base 128 — Constructive
(11, 19, 16385)-net in base 128, using
- net defined by OOA [i] based on OOA(12819, 16385, S128, 8, 8), using
- OA 4-folding and stacking [i] based on OA(12819, 65540, S128, 8), using
- discarding factors based on OA(12819, 65541, S128, 8), using
- discarding parts of the base [i] based on linear OA(25616, 65541, F256, 8) (dual of [65541, 65525, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(25615, 65536, F256, 8) (dual of [65536, 65521, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(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 Ce(7) ⊂ Ce(5) [i] based on
- discarding parts of the base [i] based on linear OA(25616, 65541, F256, 8) (dual of [65541, 65525, 9]-code), using
- discarding factors based on OA(12819, 65541, S128, 8), using
- OA 4-folding and stacking [i] based on OA(12819, 65540, S128, 8), using
(11, 11+8, 16515)-Net over F128 — Digital
Digital (11, 19, 16515)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12819, 16515, F128, 8) (dual of [16515, 16496, 9]-code), using
- (u, u+v)-construction [i] based on
- linear OA(1284, 129, F128, 4) (dual of [129, 125, 5]-code or 129-arc in PG(3,128)), using
- extended Reed–Solomon code RSe(125,128) [i]
- linear OA(12815, 16386, F128, 8) (dual of [16386, 16371, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(12815, 16384, F128, 8) (dual of [16384, 16369, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(12813, 16384, F128, 7) (dual of [16384, 16371, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(1280, 2, F128, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(1284, 129, F128, 4) (dual of [129, 125, 5]-code or 129-arc in PG(3,128)), using
- (u, u+v)-construction [i] based on
(11, 11+8, 32769)-Net in Base 128
(11, 19, 32769)-net in base 128, using
- net defined by OOA [i] based on OOA(12819, 32769, S128, 12, 8), using
- OOA stacking with additional row [i] based on OOA(12819, 32770, S128, 4, 8), using
- discarding parts of the base [i] based on linear OOA(25616, 32770, F256, 4, 8) (dual of [(32770, 4), 131064, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25616, 32770, F256, 2, 8) (dual of [(32770, 2), 65524, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25616, 65540, F256, 8) (dual of [65540, 65524, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(25616, 65541, F256, 8) (dual of [65541, 65525, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(25615, 65536, F256, 8) (dual of [65536, 65521, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(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 Ce(7) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(25616, 65541, F256, 8) (dual of [65541, 65525, 9]-code), using
- OOA 2-folding [i] based on linear OA(25616, 65540, F256, 8) (dual of [65540, 65524, 9]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25616, 32770, F256, 2, 8) (dual of [(32770, 2), 65524, 9]-NRT-code), using
- discarding parts of the base [i] based on linear OOA(25616, 32770, F256, 4, 8) (dual of [(32770, 4), 131064, 9]-NRT-code), using
- OOA stacking with additional row [i] based on OOA(12819, 32770, S128, 4, 8), using
(11, 11+8, large)-Net in Base 128 — Upper bound on s
There is no (11, 19, large)-net in base 128, because
- 6 times m-reduction [i] would yield (11, 13, large)-net in base 128, but