Best Known (45, 45+16, s)-Nets in Base 5
(45, 45+16, 390)-Net over F5 — Constructive and digital
Digital (45, 61, 390)-net over F5, using
- net defined by OOA [i] based on linear OOA(561, 390, F5, 16, 16) (dual of [(390, 16), 6179, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(561, 3120, F5, 16) (dual of [3120, 3059, 17]-code), using
- discarding factors / shortening the dual code based on 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]
- discarding factors / shortening the dual code based on linear OA(561, 3125, F5, 16) (dual of [3125, 3064, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(561, 3120, F5, 16) (dual of [3120, 3059, 17]-code), using
(45, 45+16, 1562)-Net over F5 — Digital
Digital (45, 61, 1562)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(561, 1562, F5, 2, 16) (dual of [(1562, 2), 3063, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(561, 3124, F5, 16) (dual of [3124, 3063, 17]-code), using
- discarding factors / shortening the dual code based on 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]
- discarding factors / shortening the dual code based on linear OA(561, 3125, F5, 16) (dual of [3125, 3064, 17]-code), using
- OOA 2-folding [i] based on linear OA(561, 3124, F5, 16) (dual of [3124, 3063, 17]-code), using
(45, 45+16, 201031)-Net in Base 5 — Upper bound on s
There is no (45, 61, 201032)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 4 336866 022556 388982 766459 151459 707768 679425 > 561 [i]