Best Known (122, 122+22, s)-Nets in Base 4
(122, 122+22, 5973)-Net over F4 — Constructive and digital
Digital (122, 144, 5973)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (4, 15, 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 (107, 129, 5958)-net over F4, using
- net defined by OOA [i] based on linear OOA(4129, 5958, F4, 22, 22) (dual of [(5958, 22), 130947, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(4129, 65538, F4, 22) (dual of [65538, 65409, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4129, 65544, F4, 22) (dual of [65544, 65415, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- linear OA(4129, 65536, F4, 22) (dual of [65536, 65407, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(4121, 65536, F4, 21) (dual of [65536, 65415, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(40, 8, F4, 0) (dual of [8, 8, 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(21) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(4129, 65544, F4, 22) (dual of [65544, 65415, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(4129, 65538, F4, 22) (dual of [65538, 65409, 23]-code), using
- net defined by OOA [i] based on linear OOA(4129, 5958, F4, 22, 22) (dual of [(5958, 22), 130947, 23]-NRT-code), using
- digital (4, 15, 15)-net over F4, using
(122, 122+22, 55820)-Net over F4 — Digital
Digital (122, 144, 55820)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4144, 55820, F4, 22) (dual of [55820, 55676, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4144, 65552, F4, 22) (dual of [65552, 65408, 23]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(47, 8, F4, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,4)), using
- dual of repetition code with length 8 [i]
- linear OA(48, 8, F4, 8) (dual of [8, 0, 9]-code or 8-arc in PG(7,4)), using
- linear OA(4129, 65536, F4, 22) (dual of [65536, 65407, 23]-code), using
- an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(47, 8, F4, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,4)), using
- (u, u−v, u+v+w)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(4144, 65552, F4, 22) (dual of [65552, 65408, 23]-code), using
(122, 122+22, large)-Net in Base 4 — Upper bound on s
There is no (122, 144, large)-net in base 4, because
- 20 times m-reduction [i] would yield (122, 124, large)-net in base 4, but