Best Known (84, 84+13, s)-Nets in Base 5
(84, 84+13, 325528)-Net over F5 — Constructive and digital
Digital (84, 97, 325528)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 6, 6)-net over F5, using
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 0 and N(F) ≥ 6, using
- the rational function field F5(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- digital (78, 91, 325522)-net over F5, using
- net defined by OOA [i] based on linear OOA(591, 325522, F5, 13, 13) (dual of [(325522, 13), 4231695, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(591, 1953133, F5, 13) (dual of [1953133, 1953042, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(591, 1953134, F5, 13) (dual of [1953134, 1953043, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(591, 1953125, F5, 13) (dual of [1953125, 1953034, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(582, 1953125, F5, 12) (dual of [1953125, 1953043, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(50, 9, F5, 0) (dual of [9, 9, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(591, 1953134, F5, 13) (dual of [1953134, 1953043, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(591, 1953133, F5, 13) (dual of [1953133, 1953042, 14]-code), using
- net defined by OOA [i] based on linear OOA(591, 325522, F5, 13, 13) (dual of [(325522, 13), 4231695, 14]-NRT-code), using
- digital (0, 6, 6)-net over F5, using
(84, 84+13, 1545447)-Net over F5 — Digital
Digital (84, 97, 1545447)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(597, 1545447, F5, 13) (dual of [1545447, 1545350, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(597, 1953133, F5, 13) (dual of [1953133, 1953036, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([1,6]) [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(590, 1953126, F5, 6) (dual of [1953126, 1953036, 7]-code), using the narrow-sense BCH-code C(I) with length 1953126 | 518−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(56, 7, F5, 6) (dual of [7, 1, 7]-code or 7-arc in PG(5,5)), using
- dual of repetition code with length 7 [i]
- construction X applied to C([0,6]) ⊂ C([1,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(597, 1953133, F5, 13) (dual of [1953133, 1953036, 14]-code), using
(84, 84+13, large)-Net in Base 5 — Upper bound on s
There is no (84, 97, large)-net in base 5, because
- 11 times m-reduction [i] would yield (84, 86, large)-net in base 5, but