Best Known (61, 78, s)-Nets in Base 5
(61, 78, 488)-Net over F5 — Constructive and digital
Digital (61, 78, 488)-net over F5, using
- net defined by OOA [i] based on linear OOA(578, 488, F5, 17, 17) (dual of [(488, 17), 8218, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(578, 3905, F5, 17) (dual of [3905, 3827, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(578, 3906, F5, 17) (dual of [3906, 3828, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(578, 3905, F5, 17) (dual of [3905, 3827, 18]-code), using
(61, 78, 4375)-Net over F5 — Digital
Digital (61, 78, 4375)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(578, 4375, F5, 17) (dual of [4375, 4297, 18]-code), using
- 1233 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 21 times 0, 1, 44 times 0, 1, 85 times 0, 1, 147 times 0, 1, 225 times 0, 1, 308 times 0, 1, 381 times 0) [i] based on linear OA(566, 3130, F5, 17) (dual of [3130, 3064, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(566, 3125, F5, 17) (dual of [3125, 3059, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(561, 3125, F5, 16) (dual of [3125, 3064, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(50, 5, F5, 0) (dual of [5, 5, 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
- 1233 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 21 times 0, 1, 44 times 0, 1, 85 times 0, 1, 147 times 0, 1, 225 times 0, 1, 308 times 0, 1, 381 times 0) [i] based on linear OA(566, 3130, F5, 17) (dual of [3130, 3064, 18]-code), using
(61, 78, 5025923)-Net in Base 5 — Upper bound on s
There is no (61, 78, 5025924)-net in base 5, because
- 1 times m-reduction [i] would yield (61, 77, 5025924)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 661744 813111 829056 793225 110675 093990 803288 510414 486529 > 577 [i]