Best Known (80, 80+13, s)-Nets in Base 5
(80, 80+13, 325524)-Net over F5 — Constructive and digital
Digital (80, 93, 325524)-net over F5, using
- 51 times duplication [i] based on digital (79, 92, 325524)-net over F5, using
- net defined by OOA [i] based on linear OOA(592, 325524, F5, 13, 13) (dual of [(325524, 13), 4231720, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(592, 1953145, F5, 13) (dual of [1953145, 1953053, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(591, 1953126, F5, 13) (dual of [1953126, 1953035, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 1953126 | 518−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(573, 1953126, F5, 11) (dual of [1953126, 1953053, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 1953126 | 518−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(51, 19, F5, 1) (dual of [19, 18, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(592, 1953145, F5, 13) (dual of [1953145, 1953053, 14]-code), using
- net defined by OOA [i] based on linear OOA(592, 325524, F5, 13, 13) (dual of [(325524, 13), 4231720, 14]-NRT-code), using
(80, 80+13, 976573)-Net over F5 — Digital
Digital (80, 93, 976573)-net over F5, using
- 51 times duplication [i] based on digital (79, 92, 976573)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(592, 976573, F5, 2, 13) (dual of [(976573, 2), 1953054, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(592, 1953146, F5, 13) (dual of [1953146, 1953054, 14]-code), using
- construction X4 applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(591, 1953126, F5, 13) (dual of [1953126, 1953035, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 1953126 | 518−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(573, 1953126, F5, 11) (dual of [1953126, 1953053, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 1953126 | 518−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(519, 20, F5, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,5)), using
- dual of repetition code with length 20 [i]
- linear OA(51, 20, F5, 1) (dual of [20, 19, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to C([0,6]) ⊂ C([0,5]) [i] based on
- OOA 2-folding [i] based on linear OA(592, 1953146, F5, 13) (dual of [1953146, 1953054, 14]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(592, 976573, F5, 2, 13) (dual of [(976573, 2), 1953054, 14]-NRT-code), using
(80, 80+13, large)-Net in Base 5 — Upper bound on s
There is no (80, 93, large)-net in base 5, because
- 11 times m-reduction [i] would yield (80, 82, large)-net in base 5, but