Best Known (163, 163+26, s)-Nets in Base 4
(163, 163+26, 20180)-Net over F4 — Constructive and digital
Digital (163, 189, 20180)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (4, 17, 15)-net over F4, using
- net from sequence [i] based on digital (4, 14)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 4 and N(F) ≥ 15, using
- net from sequence [i] based on digital (4, 14)-sequence over F4, using
- digital (146, 172, 20165)-net over F4, using
- net defined by OOA [i] based on linear OOA(4172, 20165, F4, 26, 26) (dual of [(20165, 26), 524118, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(4172, 262145, F4, 26) (dual of [262145, 261973, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(4172, 262153, F4, 26) (dual of [262153, 261981, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- linear OA(4172, 262144, F4, 26) (dual of [262144, 261972, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(4163, 262144, F4, 25) (dual of [262144, 261981, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(40, 9, F4, 0) (dual of [9, 9, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(4172, 262153, F4, 26) (dual of [262153, 261981, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(4172, 262145, F4, 26) (dual of [262145, 261973, 27]-code), using
- net defined by OOA [i] based on linear OOA(4172, 20165, F4, 26, 26) (dual of [(20165, 26), 524118, 27]-NRT-code), using
- digital (4, 17, 15)-net over F4, using
(163, 163+26, 169953)-Net over F4 — Digital
Digital (163, 189, 169953)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4189, 169953, F4, 26) (dual of [169953, 169764, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(4189, 262162, F4, 26) (dual of [262162, 261973, 27]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(48, 9, F4, 8) (dual of [9, 1, 9]-code or 9-arc in PG(7,4)), using
- dual of repetition code with length 9 [i]
- linear OA(49, 9, F4, 9) (dual of [9, 0, 10]-code or 9-arc in PG(8,4)), using
- linear OA(4172, 262144, F4, 26) (dual of [262144, 261972, 27]-code), using
- an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(48, 9, F4, 8) (dual of [9, 1, 9]-code or 9-arc in PG(7,4)), using
- (u, u−v, u+v+w)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(4189, 262162, F4, 26) (dual of [262162, 261973, 27]-code), using
(163, 163+26, large)-Net in Base 4 — Upper bound on s
There is no (163, 189, large)-net in base 4, because
- 24 times m-reduction [i] would yield (163, 165, large)-net in base 4, but