Best Known (85, 85+13, s)-Nets in Base 5
(85, 85+13, 325532)-Net over F5 — Constructive and digital
Digital (85, 98, 325532)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 10)-net over F5, using
- net from sequence [i] based on digital (1, 9)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 1 and N(F) ≥ 10, using
- net from sequence [i] based on digital (1, 9)-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 (1, 7, 10)-net over F5, using
(85, 85+13, 1788945)-Net over F5 — Digital
Digital (85, 98, 1788945)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(598, 1788945, F5, 13) (dual of [1788945, 1788847, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(598, 1953136, F5, 13) (dual of [1953136, 1953038, 14]-code), using
- (u, u+v)-construction [i] based on
- linear OA(57, 10, F5, 6) (dual of [10, 3, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(57, 11, F5, 6) (dual of [11, 4, 7]-code), using
- 1 times truncation [i] based on linear OA(58, 12, F5, 7) (dual of [12, 4, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(57, 11, F5, 6) (dual of [11, 4, 7]-code), using
- 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(57, 10, F5, 6) (dual of [10, 3, 7]-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(598, 1953136, F5, 13) (dual of [1953136, 1953038, 14]-code), using
(85, 85+13, large)-Net in Base 5 — Upper bound on s
There is no (85, 98, large)-net in base 5, because
- 11 times m-reduction [i] would yield (85, 87, large)-net in base 5, but