Best Known (140−19, 140, s)-Nets in Base 5
(140−19, 140, 217017)-Net over F5 — Constructive and digital
Digital (121, 140, 217017)-net over F5, using
- 51 times duplication [i] based on digital (120, 139, 217017)-net over F5, using
- net defined by OOA [i] based on linear OOA(5139, 217017, F5, 19, 19) (dual of [(217017, 19), 4123184, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(5139, 1953154, F5, 19) (dual of [1953154, 1953015, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(5139, 1953155, F5, 19) (dual of [1953155, 1953016, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(5136, 1953125, F5, 19) (dual of [1953125, 1952989, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(5109, 1953125, F5, 16) (dual of [1953125, 1953016, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(53, 30, F5, 2) (dual of [30, 27, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(53, 31, F5, 2) (dual of [31, 28, 3]-code), using
- Hamming code H(3,5) [i]
- discarding factors / shortening the dual code based on linear OA(53, 31, F5, 2) (dual of [31, 28, 3]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(5139, 1953155, F5, 19) (dual of [1953155, 1953016, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(5139, 1953154, F5, 19) (dual of [1953154, 1953015, 20]-code), using
- net defined by OOA [i] based on linear OOA(5139, 217017, F5, 19, 19) (dual of [(217017, 19), 4123184, 20]-NRT-code), using
(140−19, 140, 976578)-Net over F5 — Digital
Digital (121, 140, 976578)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(5140, 976578, F5, 2, 19) (dual of [(976578, 2), 1953016, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(5140, 1953156, F5, 19) (dual of [1953156, 1953016, 20]-code), using
- 1 times code embedding in larger space [i] based on linear OA(5139, 1953155, F5, 19) (dual of [1953155, 1953016, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(5136, 1953125, F5, 19) (dual of [1953125, 1952989, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(5109, 1953125, F5, 16) (dual of [1953125, 1953016, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(53, 30, F5, 2) (dual of [30, 27, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(53, 31, F5, 2) (dual of [31, 28, 3]-code), using
- Hamming code H(3,5) [i]
- discarding factors / shortening the dual code based on linear OA(53, 31, F5, 2) (dual of [31, 28, 3]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(5139, 1953155, F5, 19) (dual of [1953155, 1953016, 20]-code), using
- OOA 2-folding [i] based on linear OA(5140, 1953156, F5, 19) (dual of [1953156, 1953016, 20]-code), using
(140−19, 140, large)-Net in Base 5 — Upper bound on s
There is no (121, 140, large)-net in base 5, because
- 17 times m-reduction [i] would yield (121, 123, large)-net in base 5, but