Best Known (22−5, 22, s)-Nets in Base 4
(22−5, 22, 8191)-Net over F4 — Constructive and digital
Digital (17, 22, 8191)-net over F4, using
- net defined by OOA [i] based on linear OOA(422, 8191, F4, 5, 5) (dual of [(8191, 5), 40933, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(422, 16383, F4, 5) (dual of [16383, 16361, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(422, 16384, F4, 5) (dual of [16384, 16362, 6]-code), using
- an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(422, 16384, F4, 5) (dual of [16384, 16362, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(422, 16383, F4, 5) (dual of [16383, 16361, 6]-code), using
(22−5, 22, 9922)-Net over F4 — Digital
Digital (17, 22, 9922)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(422, 9922, F4, 5) (dual of [9922, 9900, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(422, 16384, F4, 5) (dual of [16384, 16362, 6]-code), using
- an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(422, 16384, F4, 5) (dual of [16384, 16362, 6]-code), using
(22−5, 22, 988605)-Net in Base 4 — Upper bound on s
There is no (17, 22, 988606)-net in base 4, because
- 1 times m-reduction [i] would yield (17, 21, 988606)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 4 398048 584926 > 421 [i]