Best Known (125−17, 125, s)-Nets in Base 5
(125−17, 125, 244145)-Net over F5 — Constructive and digital
Digital (108, 125, 244145)-net over F5, using
- net defined by OOA [i] based on linear OOA(5125, 244145, F5, 17, 17) (dual of [(244145, 17), 4150340, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(5125, 1953161, F5, 17) (dual of [1953161, 1953036, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(5125, 1953168, F5, 17) (dual of [1953168, 1953043, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(11) [i] based on
- linear OA(5118, 1953125, F5, 17) (dual of [1953125, 1953007, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 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(57, 43, F5, 4) (dual of [43, 36, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(57, 44, F5, 4) (dual of [44, 37, 5]-code), using
- construction X applied to Ce(16) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(5125, 1953168, F5, 17) (dual of [1953168, 1953043, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(5125, 1953161, F5, 17) (dual of [1953161, 1953036, 18]-code), using
(125−17, 125, 976584)-Net over F5 — Digital
Digital (108, 125, 976584)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(5125, 976584, F5, 2, 17) (dual of [(976584, 2), 1953043, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(5125, 1953168, F5, 17) (dual of [1953168, 1953043, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(11) [i] based on
- linear OA(5118, 1953125, F5, 17) (dual of [1953125, 1953007, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 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(57, 43, F5, 4) (dual of [43, 36, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(57, 44, F5, 4) (dual of [44, 37, 5]-code), using
- construction X applied to Ce(16) ⊂ Ce(11) [i] based on
- OOA 2-folding [i] based on linear OA(5125, 1953168, F5, 17) (dual of [1953168, 1953043, 18]-code), using
(125−17, 125, large)-Net in Base 5 — Upper bound on s
There is no (108, 125, large)-net in base 5, because
- 15 times m-reduction [i] would yield (108, 110, large)-net in base 5, but