Best Known (134−27, 134, s)-Nets in Base 5
(134−27, 134, 1204)-Net over F5 — Constructive and digital
Digital (107, 134, 1204)-net over F5, using
- 51 times duplication [i] based on digital (106, 133, 1204)-net over F5, using
- net defined by OOA [i] based on linear OOA(5133, 1204, F5, 27, 27) (dual of [(1204, 27), 32375, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(5133, 15653, F5, 27) (dual of [15653, 15520, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(5133, 15655, F5, 27) (dual of [15655, 15522, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(21) [i] based on
- linear OA(5127, 15625, F5, 27) (dual of [15625, 15498, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(5103, 15625, F5, 22) (dual of [15625, 15522, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(56, 30, F5, 4) (dual of [30, 24, 5]-code), using
- construction X applied to Ce(26) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(5133, 15655, F5, 27) (dual of [15655, 15522, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(5133, 15653, F5, 27) (dual of [15653, 15520, 28]-code), using
- net defined by OOA [i] based on linear OOA(5133, 1204, F5, 27, 27) (dual of [(1204, 27), 32375, 28]-NRT-code), using
(134−27, 134, 13290)-Net over F5 — Digital
Digital (107, 134, 13290)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5134, 13290, F5, 27) (dual of [13290, 13156, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(5134, 15639, F5, 27) (dual of [15639, 15505, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(5133, 15626, F5, 27) (dual of [15626, 15493, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(5121, 15626, F5, 25) (dual of [15626, 15505, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [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 C([0,13]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(5134, 15639, F5, 27) (dual of [15639, 15505, 28]-code), using
(134−27, 134, large)-Net in Base 5 — Upper bound on s
There is no (107, 134, large)-net in base 5, because
- 25 times m-reduction [i] would yield (107, 109, large)-net in base 5, but