Best Known (93−19, 93, s)-Nets in Base 5
(93−19, 93, 1737)-Net over F5 — Constructive and digital
Digital (74, 93, 1737)-net over F5, using
- 51 times duplication [i] based on digital (73, 92, 1737)-net over F5, using
- net defined by OOA [i] based on linear OOA(592, 1737, F5, 19, 19) (dual of [(1737, 19), 32911, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(592, 15634, F5, 19) (dual of [15634, 15542, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(592, 15638, F5, 19) (dual of [15638, 15546, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(591, 15625, F5, 19) (dual of [15625, 15534, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(579, 15625, F5, 17) (dual of [15625, 15546, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(51, 13, F5, 1) (dual of [13, 12, 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 Ce(18) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(592, 15638, F5, 19) (dual of [15638, 15546, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(592, 15634, F5, 19) (dual of [15634, 15542, 20]-code), using
- net defined by OOA [i] based on linear OOA(592, 1737, F5, 19, 19) (dual of [(1737, 19), 32911, 20]-NRT-code), using
(93−19, 93, 10866)-Net over F5 — Digital
Digital (74, 93, 10866)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(593, 10866, F5, 19) (dual of [10866, 10773, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(593, 15640, F5, 19) (dual of [15640, 15547, 20]-code), using
- construction XX applied to Ce(18) ⊂ Ce(16) ⊂ Ce(15) [i] based on
- linear OA(591, 15625, F5, 19) (dual of [15625, 15534, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(579, 15625, F5, 17) (dual of [15625, 15546, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(573, 15625, F5, 16) (dual of [15625, 15552, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(51, 14, F5, 1) (dual of [14, 13, 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
- linear OA(50, 1, F5, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(18) ⊂ Ce(16) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(593, 15640, F5, 19) (dual of [15640, 15547, 20]-code), using
(93−19, 93, large)-Net in Base 5 — Upper bound on s
There is no (74, 93, large)-net in base 5, because
- 17 times m-reduction [i] would yield (74, 76, large)-net in base 5, but