Best Known (132, 142, s)-Nets in Base 4
(132, 142, 6841957)-Net over F4 — Constructive and digital
Digital (132, 142, 6841957)-net over F4, using
- 41 times duplication [i] based on digital (131, 141, 6841957)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (24, 29, 131077)-net over F4, using
- net defined by OOA [i] based on linear OOA(429, 131077, F4, 5, 5) (dual of [(131077, 5), 655356, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(429, 262155, F4, 5) (dual of [262155, 262126, 6]-code), using
- construction X4 applied to Ce(4) ⊂ Ce(2) [i] based on
- linear OA(428, 262144, F4, 5) (dual of [262144, 262116, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(419, 262144, F4, 3) (dual of [262144, 262125, 4]-code or 262144-cap in PG(18,4)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(410, 11, F4, 10) (dual of [11, 1, 11]-code or 11-arc in PG(9,4)), using
- dual of repetition code with length 11 [i]
- linear OA(41, 11, F4, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(4) ⊂ Ce(2) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(429, 262155, F4, 5) (dual of [262155, 262126, 6]-code), using
- net defined by OOA [i] based on linear OOA(429, 131077, F4, 5, 5) (dual of [(131077, 5), 655356, 6]-NRT-code), using
- digital (102, 112, 6710880)-net over F4, using
- trace code for nets [i] based on digital (18, 28, 1677720)-net over F256, using
- net defined by OOA [i] based on linear OOA(25628, 1677720, F256, 10, 10) (dual of [(1677720, 10), 16777172, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(25628, 8388600, F256, 10) (dual of [8388600, 8388572, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(25628, large, F256, 10) (dual of [large, large−28, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(25628, large, F256, 10) (dual of [large, large−28, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(25628, 8388600, F256, 10) (dual of [8388600, 8388572, 11]-code), using
- net defined by OOA [i] based on linear OOA(25628, 1677720, F256, 10, 10) (dual of [(1677720, 10), 16777172, 11]-NRT-code), using
- trace code for nets [i] based on digital (18, 28, 1677720)-net over F256, using
- digital (24, 29, 131077)-net over F4, using
- (u, u+v)-construction [i] based on
(132, 142, large)-Net over F4 — Digital
Digital (132, 142, large)-net over F4, using
- t-expansion [i] based on digital (130, 142, large)-net over F4, using
- 3 times m-reduction [i] based on digital (130, 145, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4145, large, F4, 15) (dual of [large, large−145, 16]-code), using
- strength reduction [i] based on linear OA(4145, large, F4, 17) (dual of [large, large−145, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 424−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- strength reduction [i] based on linear OA(4145, large, F4, 17) (dual of [large, large−145, 18]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4145, large, F4, 15) (dual of [large, large−145, 16]-code), using
- 3 times m-reduction [i] based on digital (130, 145, large)-net over F4, using
(132, 142, large)-Net in Base 4 — Upper bound on s
There is no (132, 142, large)-net in base 4, because
- 8 times m-reduction [i] would yield (132, 134, large)-net in base 4, but