Best Known (55, 55+14, s)-Nets in Base 5
(55, 55+14, 2234)-Net over F5 — Constructive and digital
Digital (55, 69, 2234)-net over F5, using
- 51 times duplication [i] based on digital (54, 68, 2234)-net over F5, using
- net defined by OOA [i] based on linear OOA(568, 2234, F5, 14, 14) (dual of [(2234, 14), 31208, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(568, 15638, F5, 14) (dual of [15638, 15570, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(567, 15625, F5, 14) (dual of [15625, 15558, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(555, 15625, F5, 12) (dual of [15625, 15570, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(13) ⊂ Ce(11) [i] based on
- OA 7-folding and stacking [i] based on linear OA(568, 15638, F5, 14) (dual of [15638, 15570, 15]-code), using
- net defined by OOA [i] based on linear OOA(568, 2234, F5, 14, 14) (dual of [(2234, 14), 31208, 15]-NRT-code), using
(55, 55+14, 12074)-Net over F5 — Digital
Digital (55, 69, 12074)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(569, 12074, F5, 14) (dual of [12074, 12005, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(569, 15640, F5, 14) (dual of [15640, 15571, 15]-code), using
- construction XX applied to Ce(13) ⊂ Ce(11) ⊂ Ce(10) [i] based on
- linear OA(567, 15625, F5, 14) (dual of [15625, 15558, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(555, 15625, F5, 12) (dual of [15625, 15570, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(549, 15625, F5, 11) (dual of [15625, 15576, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(13) ⊂ Ce(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(569, 15640, F5, 14) (dual of [15640, 15571, 15]-code), using
(55, 55+14, 6557005)-Net in Base 5 — Upper bound on s
There is no (55, 69, 6557006)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 1 694067 351092 958607 329442 079847 707014 991220 699225 > 569 [i]