Best Known (15, 15+11, s)-Nets in Base 128
(15, 15+11, 3406)-Net over F128 — Constructive and digital
Digital (15, 26, 3406)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (0, 5, 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 (10, 21, 3277)-net over F128, using
- net defined by OOA [i] based on linear OOA(12821, 3277, F128, 11, 11) (dual of [(3277, 11), 36026, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(12821, 16386, F128, 11) (dual of [16386, 16365, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(12821, 16384, F128, 11) (dual of [16384, 16363, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(12819, 16384, F128, 10) (dual of [16384, 16365, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(10) ⊂ Ce(9) [i] based on
- OOA 5-folding and stacking with additional row [i] based on linear OA(12821, 16386, F128, 11) (dual of [16386, 16365, 12]-code), using
- net defined by OOA [i] based on linear OOA(12821, 3277, F128, 11, 11) (dual of [(3277, 11), 36026, 12]-NRT-code), using
- digital (0, 5, 129)-net over F128, using
(15, 15+11, 13108)-Net in Base 128 — Constructive
(15, 26, 13108)-net in base 128, using
- net defined by OOA [i] based on OOA(12826, 13108, S128, 11, 11), using
- OOA 5-folding and stacking with additional row [i] based on OA(12826, 65541, S128, 11), using
- discarding factors based on OA(12826, 65542, S128, 11), using
- discarding parts of the base [i] based on linear OA(25622, 65542, F256, 11) (dual of [65542, 65520, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- linear OA(25621, 65537, F256, 11) (dual of [65537, 65516, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- 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(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([0,5]) ⊂ C([0,4]) [i] based on
- discarding parts of the base [i] based on linear OA(25622, 65542, F256, 11) (dual of [65542, 65520, 12]-code), using
- discarding factors based on OA(12826, 65542, S128, 11), using
- OOA 5-folding and stacking with additional row [i] based on OA(12826, 65541, S128, 11), using
(15, 15+11, 16515)-Net over F128 — Digital
Digital (15, 26, 16515)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12826, 16515, F128, 11) (dual of [16515, 16489, 12]-code), using
- (u, u+v)-construction [i] based on
- linear OA(1285, 129, F128, 5) (dual of [129, 124, 6]-code or 129-arc in PG(4,128)), using
- extended Reed–Solomon code RSe(124,128) [i]
- linear OA(12821, 16386, F128, 11) (dual of [16386, 16365, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(12821, 16384, F128, 11) (dual of [16384, 16363, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(12819, 16384, F128, 10) (dual of [16384, 16365, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(10) ⊂ Ce(9) [i] based on
- linear OA(1285, 129, F128, 5) (dual of [129, 124, 6]-code or 129-arc in PG(4,128)), using
- (u, u+v)-construction [i] based on
(15, 15+11, 20651)-Net in Base 128
(15, 26, 20651)-net in base 128, using
- 1282 times duplication [i] based on (13, 24, 20651)-net in base 128, using
- base change [i] based on digital (10, 21, 20651)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25621, 20651, F256, 3, 11) (dual of [(20651, 3), 61932, 12]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(25621, 21846, F256, 3, 11) (dual of [(21846, 3), 65517, 12]-NRT-code), using
- OOA 3-folding [i] based on linear OA(25621, 65538, F256, 11) (dual of [65538, 65517, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(25621, 65536, F256, 11) (dual of [65536, 65515, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(25619, 65536, F256, 10) (dual of [65536, 65517, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- OOA 3-folding [i] based on linear OA(25621, 65538, F256, 11) (dual of [65538, 65517, 12]-code), using
- discarding factors / shortening the dual code based on linear OOA(25621, 21846, F256, 3, 11) (dual of [(21846, 3), 65517, 12]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25621, 20651, F256, 3, 11) (dual of [(20651, 3), 61932, 12]-NRT-code), using
- base change [i] based on digital (10, 21, 20651)-net over F256, using
(15, 15+11, large)-Net in Base 128 — Upper bound on s
There is no (15, 26, large)-net in base 128, because
- 9 times m-reduction [i] would yield (15, 17, large)-net in base 128, but