Best Known (56, 67, s)-Nets in Base 5
(56, 67, 78127)-Net over F5 — Constructive and digital
Digital (56, 67, 78127)-net over F5, using
- net defined by OOA [i] based on linear OOA(567, 78127, F5, 11, 11) (dual of [(78127, 11), 859330, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(567, 390636, F5, 11) (dual of [390636, 390569, 12]-code), using
- construction XX applied to Ce(10) ⊂ Ce(8) ⊂ Ce(7) [i] based on
- 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(557, 390625, F5, 9) (dual of [390625, 390568, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(549, 390625, F5, 8) (dual of [390625, 390576, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(51, 10, F5, 1) (dual of [10, 9, 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(10) ⊂ Ce(8) ⊂ Ce(7) [i] based on
- OOA 5-folding and stacking with additional row [i] based on linear OA(567, 390636, F5, 11) (dual of [390636, 390569, 12]-code), using
(56, 67, 195318)-Net over F5 — Digital
Digital (56, 67, 195318)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(567, 195318, F5, 2, 11) (dual of [(195318, 2), 390569, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(567, 390636, F5, 11) (dual of [390636, 390569, 12]-code), using
- construction XX applied to Ce(10) ⊂ Ce(8) ⊂ Ce(7) [i] based on
- 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(557, 390625, F5, 9) (dual of [390625, 390568, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(549, 390625, F5, 8) (dual of [390625, 390576, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(51, 10, F5, 1) (dual of [10, 9, 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(10) ⊂ Ce(8) ⊂ Ce(7) [i] based on
- OOA 2-folding [i] based on linear OA(567, 390636, F5, 11) (dual of [390636, 390569, 12]-code), using
(56, 67, large)-Net in Base 5 — Upper bound on s
There is no (56, 67, large)-net in base 5, because
- 9 times m-reduction [i] would yield (56, 58, large)-net in base 5, but