Best Known (123−17, 123, s)-Nets in Base 5
(123−17, 123, 244144)-Net over F5 — Constructive and digital
Digital (106, 123, 244144)-net over F5, using
- net defined by OOA [i] based on linear OOA(5123, 244144, F5, 17, 17) (dual of [(244144, 17), 4150325, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(5123, 1953153, F5, 17) (dual of [1953153, 1953030, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(5123, 1953157, F5, 17) (dual of [1953157, 1953034, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- linear OA(5118, 1953125, F5, 17) (dual of [1953125, 1953007, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 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(55, 32, F5, 3) (dual of [32, 27, 4]-code or 32-cap in PG(4,5)), using
- discarding factors / shortening the dual code based on linear OA(55, 42, F5, 3) (dual of [42, 37, 4]-code or 42-cap in PG(4,5)), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(5123, 1953157, F5, 17) (dual of [1953157, 1953034, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(5123, 1953153, F5, 17) (dual of [1953153, 1953030, 18]-code), using
(123−17, 123, 976578)-Net over F5 — Digital
Digital (106, 123, 976578)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(5123, 976578, F5, 2, 17) (dual of [(976578, 2), 1953033, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(5123, 1953156, F5, 17) (dual of [1953156, 1953033, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(5123, 1953157, F5, 17) (dual of [1953157, 1953034, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- linear OA(5118, 1953125, F5, 17) (dual of [1953125, 1953007, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 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(55, 32, F5, 3) (dual of [32, 27, 4]-code or 32-cap in PG(4,5)), using
- discarding factors / shortening the dual code based on linear OA(55, 42, F5, 3) (dual of [42, 37, 4]-code or 42-cap in PG(4,5)), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(5123, 1953157, F5, 17) (dual of [1953157, 1953034, 18]-code), using
- OOA 2-folding [i] based on linear OA(5123, 1953156, F5, 17) (dual of [1953156, 1953033, 18]-code), using
(123−17, 123, large)-Net in Base 5 — Upper bound on s
There is no (106, 123, large)-net in base 5, because
- 15 times m-reduction [i] would yield (106, 108, large)-net in base 5, but