Best Known (82−12, 82, s)-Nets in Base 5
(82−12, 82, 325522)-Net over F5 — Constructive and digital
Digital (70, 82, 325522)-net over F5, using
- net defined by OOA [i] based on linear OOA(582, 325522, F5, 12, 12) (dual of [(325522, 12), 3906182, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(582, 1953132, F5, 12) (dual of [1953132, 1953050, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(582, 1953134, F5, 12) (dual of [1953134, 1953052, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(582, 1953125, F5, 12) (dual of [1953125, 1953043, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(573, 1953125, F5, 11) (dual of [1953125, 1953052, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(50, 9, F5, 0) (dual of [9, 9, 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(582, 1953134, F5, 12) (dual of [1953134, 1953052, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(582, 1953132, F5, 12) (dual of [1953132, 1953050, 13]-code), using
(82−12, 82, 976567)-Net over F5 — Digital
Digital (70, 82, 976567)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(582, 976567, F5, 2, 12) (dual of [(976567, 2), 1953052, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(582, 1953134, F5, 12) (dual of [1953134, 1953052, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(582, 1953125, F5, 12) (dual of [1953125, 1953043, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(573, 1953125, F5, 11) (dual of [1953125, 1953052, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(50, 9, F5, 0) (dual of [9, 9, 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
- OOA 2-folding [i] based on linear OA(582, 1953134, F5, 12) (dual of [1953134, 1953052, 13]-code), using
(82−12, 82, large)-Net in Base 5 — Upper bound on s
There is no (70, 82, large)-net in base 5, because
- 10 times m-reduction [i] would yield (70, 72, large)-net in base 5, but