Best Known (147−21, 147, s)-Nets in Base 5
(147−21, 147, 195313)-Net over F5 — Constructive and digital
Digital (126, 147, 195313)-net over F5, using
- 51 times duplication [i] based on digital (125, 146, 195313)-net over F5, using
- net defined by OOA [i] based on linear OOA(5146, 195313, F5, 21, 21) (dual of [(195313, 21), 4101427, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(5146, 1953131, F5, 21) (dual of [1953131, 1952985, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(5146, 1953135, F5, 21) (dual of [1953135, 1952989, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(5145, 1953125, F5, 21) (dual of [1953125, 1952980, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(5136, 1953125, F5, 19) (dual of [1953125, 1952989, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(5146, 1953135, F5, 21) (dual of [1953135, 1952989, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(5146, 1953131, F5, 21) (dual of [1953131, 1952985, 22]-code), using
- net defined by OOA [i] based on linear OOA(5146, 195313, F5, 21, 21) (dual of [(195313, 21), 4101427, 22]-NRT-code), using
(147−21, 147, 806594)-Net over F5 — Digital
Digital (126, 147, 806594)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(5147, 806594, F5, 2, 21) (dual of [(806594, 2), 1613041, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(5147, 976568, F5, 2, 21) (dual of [(976568, 2), 1952989, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(5147, 1953136, F5, 21) (dual of [1953136, 1952989, 22]-code), using
- 1 times code embedding in larger space [i] based on linear OA(5146, 1953135, F5, 21) (dual of [1953135, 1952989, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(5145, 1953125, F5, 21) (dual of [1953125, 1952980, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(5136, 1953125, F5, 19) (dual of [1953125, 1952989, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(5146, 1953135, F5, 21) (dual of [1953135, 1952989, 22]-code), using
- OOA 2-folding [i] based on linear OA(5147, 1953136, F5, 21) (dual of [1953136, 1952989, 22]-code), using
- discarding factors / shortening the dual code based on linear OOA(5147, 976568, F5, 2, 21) (dual of [(976568, 2), 1952989, 22]-NRT-code), using
(147−21, 147, large)-Net in Base 5 — Upper bound on s
There is no (126, 147, large)-net in base 5, because
- 19 times m-reduction [i] would yield (126, 128, large)-net in base 5, but