Best Known (78−16, 78, s)-Nets in Base 5
(78−16, 78, 1956)-Net over F5 — Constructive and digital
Digital (62, 78, 1956)-net over F5, using
- net defined by OOA [i] based on linear OOA(578, 1956, F5, 16, 16) (dual of [(1956, 16), 31218, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(578, 15648, F5, 16) (dual of [15648, 15570, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(578, 15649, F5, 16) (dual of [15649, 15571, 17]-code), using
- construction XX applied to Ce(15) ⊂ Ce(11) ⊂ Ce(10) [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]
- linear OA(555, 15625, F5, 12) (dual of [15625, 15570, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(549, 15625, F5, 11) (dual of [15625, 15576, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(54, 23, F5, 3) (dual of [23, 19, 4]-code or 23-cap in PG(3,5)), using
- linear OA(50, 1, F5, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(15) ⊂ Ce(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(578, 15649, F5, 16) (dual of [15649, 15571, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(578, 15648, F5, 16) (dual of [15648, 15570, 17]-code), using
(78−16, 78, 10552)-Net over F5 — Digital
Digital (62, 78, 10552)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(578, 10552, F5, 16) (dual of [10552, 10474, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(578, 15649, F5, 16) (dual of [15649, 15571, 17]-code), using
- construction XX applied to Ce(15) ⊂ Ce(11) ⊂ Ce(10) [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]
- linear OA(555, 15625, F5, 12) (dual of [15625, 15570, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(549, 15625, F5, 11) (dual of [15625, 15576, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(54, 23, F5, 3) (dual of [23, 19, 4]-code or 23-cap in PG(3,5)), using
- linear OA(50, 1, F5, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(15) ⊂ Ce(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(578, 15649, F5, 16) (dual of [15649, 15571, 17]-code), using
(78−16, 78, 6145924)-Net in Base 5 — Upper bound on s
There is no (62, 78, 6145925)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 3 308724 183577 337819 729870 686183 683777 171072 493826 492641 > 578 [i]