Best Known (87, 100, s)-Nets in Base 5
(87, 100, 325549)-Net over F5 — Constructive and digital
Digital (87, 100, 325549)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (3, 9, 27)-net 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
(87, 100, 1953179)-Net over F5 — Digital
Digital (87, 100, 1953179)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5100, 1953179, F5, 13) (dual of [1953179, 1953079, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(6) [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(546, 1953125, F5, 7) (dual of [1953125, 1953079, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(59, 54, F5, 5) (dual of [54, 45, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(59, 62, F5, 5) (dual of [62, 53, 6]-code), using
- a “GraCyc†code from Grassl’s database [i]
- discarding factors / shortening the dual code based on linear OA(59, 62, F5, 5) (dual of [62, 53, 6]-code), using
- construction X applied to Ce(12) ⊂ Ce(6) [i] based on
(87, 100, large)-Net in Base 5 — Upper bound on s
There is no (87, 100, large)-net in base 5, because
- 11 times m-reduction [i] would yield (87, 89, large)-net in base 5, but