Best Known (61, 61+12, s)-Nets in Base 5
(61, 61+12, 65105)-Net over F5 — Constructive and digital
Digital (61, 73, 65105)-net over F5, using
- net defined by OOA [i] based on linear OOA(573, 65105, F5, 12, 12) (dual of [(65105, 12), 781187, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(573, 390630, F5, 12) (dual of [390630, 390557, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(573, 390633, F5, 12) (dual of [390633, 390560, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(573, 390625, F5, 12) (dual of [390625, 390552, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(565, 390625, F5, 11) (dual of [390625, 390560, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(50, 8, F5, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(573, 390633, F5, 12) (dual of [390633, 390560, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(573, 390630, F5, 12) (dual of [390630, 390557, 13]-code), using
(61, 61+12, 195316)-Net over F5 — Digital
Digital (61, 73, 195316)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(573, 195316, F5, 2, 12) (dual of [(195316, 2), 390559, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(573, 390632, F5, 12) (dual of [390632, 390559, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(573, 390633, F5, 12) (dual of [390633, 390560, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(573, 390625, F5, 12) (dual of [390625, 390552, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(565, 390625, F5, 11) (dual of [390625, 390560, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(50, 8, F5, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(573, 390633, F5, 12) (dual of [390633, 390560, 13]-code), using
- OOA 2-folding [i] based on linear OA(573, 390632, F5, 12) (dual of [390632, 390559, 13]-code), using
(61, 61+12, large)-Net in Base 5 — Upper bound on s
There is no (61, 73, large)-net in base 5, because
- 10 times m-reduction [i] would yield (61, 63, large)-net in base 5, but