Best Known (72−15, 72, s)-Nets in Base 5
(72−15, 72, 2231)-Net over F5 — Constructive and digital
Digital (57, 72, 2231)-net over F5, using
- net defined by OOA [i] based on linear OOA(572, 2231, F5, 15, 15) (dual of [(2231, 15), 33393, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(572, 15618, F5, 15) (dual of [15618, 15546, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(572, 15624, F5, 15) (dual of [15624, 15552, 16]-code), using
- 1 times truncation [i] based on linear OA(573, 15625, F5, 16) (dual of [15625, 15552, 17]-code), using
- an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- 1 times truncation [i] based on linear OA(573, 15625, F5, 16) (dual of [15625, 15552, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(572, 15624, F5, 15) (dual of [15624, 15552, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(572, 15618, F5, 15) (dual of [15618, 15546, 16]-code), using
(72−15, 72, 9298)-Net over F5 — Digital
Digital (57, 72, 9298)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(572, 9298, F5, 15) (dual of [9298, 9226, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(572, 15624, F5, 15) (dual of [15624, 15552, 16]-code), using
- 1 times truncation [i] based on linear OA(573, 15625, F5, 16) (dual of [15625, 15552, 17]-code), using
- an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- 1 times truncation [i] based on linear OA(573, 15625, F5, 16) (dual of [15625, 15552, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(572, 15624, F5, 15) (dual of [15624, 15552, 16]-code), using
(72−15, 72, large)-Net in Base 5 — Upper bound on s
There is no (57, 72, large)-net in base 5, because
- 13 times m-reduction [i] would yield (57, 59, large)-net in base 5, but