Best Known (143−19, 143, s)-Nets in Base 5
(143−19, 143, 217018)-Net over F5 — Constructive and digital
Digital (124, 143, 217018)-net over F5, using
- net defined by OOA [i] based on linear OOA(5143, 217018, F5, 19, 19) (dual of [(217018, 19), 4123199, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(5143, 1953163, F5, 19) (dual of [1953163, 1953020, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(5143, 1953168, F5, 19) (dual of [1953168, 1953025, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [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(5100, 1953125, F5, 14) (dual of [1953125, 1953025, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(57, 43, F5, 4) (dual of [43, 36, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(57, 44, F5, 4) (dual of [44, 37, 5]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(5143, 1953168, F5, 19) (dual of [1953168, 1953025, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(5143, 1953163, F5, 19) (dual of [1953163, 1953020, 20]-code), using
(143−19, 143, 1236905)-Net over F5 — Digital
Digital (124, 143, 1236905)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5143, 1236905, F5, 19) (dual of [1236905, 1236762, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(5143, 1953161, F5, 19) (dual of [1953161, 1953018, 20]-code), using
- construction XX applied to Ce(18) ⊂ Ce(15) ⊂ Ce(13) [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(5100, 1953125, F5, 14) (dual of [1953125, 1953025, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(54, 33, F5, 2) (dual of [33, 29, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(54, 124, F5, 2) (dual of [124, 120, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 124 = 53−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(54, 124, F5, 2) (dual of [124, 120, 3]-code), using
- linear OA(51, 3, F5, 1) (dual of [3, 2, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, 5, F5, 1) (dual of [5, 4, 2]-code), using
- Reed–Solomon code RS(4,5) [i]
- discarding factors / shortening the dual code based on linear OA(51, 5, F5, 1) (dual of [5, 4, 2]-code), using
- construction XX applied to Ce(18) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(5143, 1953161, F5, 19) (dual of [1953161, 1953018, 20]-code), using
(143−19, 143, large)-Net in Base 5 — Upper bound on s
There is no (124, 143, large)-net in base 5, because
- 17 times m-reduction [i] would yield (124, 126, large)-net in base 5, but