Best Known (79−17, 79, s)-Nets in Base 5
(79−17, 79, 1953)-Net over F5 — Constructive and digital
Digital (62, 79, 1953)-net over F5, using
- net defined by OOA [i] based on linear OOA(579, 1953, F5, 17, 17) (dual of [(1953, 17), 33122, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(579, 15625, F5, 17) (dual of [15625, 15546, 18]-code), using
- an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- OOA 8-folding and stacking with additional row [i] based on linear OA(579, 15625, F5, 17) (dual of [15625, 15546, 18]-code), using
(79−17, 79, 7815)-Net over F5 — Digital
Digital (62, 79, 7815)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(579, 7815, F5, 2, 17) (dual of [(7815, 2), 15551, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(579, 15630, F5, 17) (dual of [15630, 15551, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(579, 15631, F5, 17) (dual of [15631, 15552, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(579, 15625, F5, 17) (dual of [15625, 15546, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 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]
- linear OA(50, 6, F5, 0) (dual of [6, 6, 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(16) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(579, 15631, F5, 17) (dual of [15631, 15552, 18]-code), using
- OOA 2-folding [i] based on linear OA(579, 15630, F5, 17) (dual of [15630, 15551, 18]-code), using
(79−17, 79, 6145924)-Net in Base 5 — Upper bound on s
There is no (62, 79, 6145925)-net in base 5, because
- 1 times m-reduction [i] would yield (62, 78, 6145925)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 3 308724 183577 337819 729870 686183 683777 171072 493826 492641 > 578 [i]