Best Known (117−24, 117, s)-Nets in Base 5
(117−24, 117, 1303)-Net over F5 — Constructive and digital
Digital (93, 117, 1303)-net over F5, using
- 51 times duplication [i] based on digital (92, 116, 1303)-net over F5, using
- net defined by OOA [i] based on linear OOA(5116, 1303, F5, 24, 24) (dual of [(1303, 24), 31156, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(5116, 15636, F5, 24) (dual of [15636, 15520, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(5116, 15638, F5, 24) (dual of [15638, 15522, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(21) [i] based on
- linear OA(5115, 15625, F5, 24) (dual of [15625, 15510, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [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(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(23) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(5116, 15638, F5, 24) (dual of [15638, 15522, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(5116, 15636, F5, 24) (dual of [15636, 15520, 25]-code), using
- net defined by OOA [i] based on linear OOA(5116, 1303, F5, 24, 24) (dual of [(1303, 24), 31156, 25]-NRT-code), using
(117−24, 117, 10956)-Net over F5 — Digital
Digital (93, 117, 10956)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5117, 10956, F5, 24) (dual of [10956, 10839, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(5117, 15640, F5, 24) (dual of [15640, 15523, 25]-code), using
- construction XX applied to Ce(23) ⊂ Ce(21) ⊂ Ce(20) [i] based on
- linear OA(5115, 15625, F5, 24) (dual of [15625, 15510, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [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(597, 15625, F5, 21) (dual of [15625, 15528, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(23) ⊂ Ce(21) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(5117, 15640, F5, 24) (dual of [15640, 15523, 25]-code), using
(117−24, 117, large)-Net in Base 5 — Upper bound on s
There is no (93, 117, large)-net in base 5, because
- 22 times m-reduction [i] would yield (93, 95, large)-net in base 5, but