Best Known (121, 121+22, s)-Nets in Base 4
(121, 121+22, 5972)-Net over F4 — Constructive and digital
Digital (121, 143, 5972)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (3, 14, 14)-net over F4, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 3 and N(F) ≥ 14, using
- net from sequence [i] based on digital (3, 13)-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 (3, 14, 14)-net over F4, using
(121, 121+22, 52081)-Net over F4 — Digital
Digital (121, 143, 52081)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4143, 52081, F4, 22) (dual of [52081, 51938, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4143, 65590, F4, 22) (dual of [65590, 65447, 23]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4142, 65589, F4, 22) (dual of [65589, 65447, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(14) [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(489, 65536, F4, 15) (dual of [65536, 65447, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(413, 53, F4, 6) (dual of [53, 40, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- construction X applied to Ce(21) ⊂ Ce(14) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4142, 65589, F4, 22) (dual of [65589, 65447, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4143, 65590, F4, 22) (dual of [65590, 65447, 23]-code), using
(121, 121+22, large)-Net in Base 4 — Upper bound on s
There is no (121, 143, large)-net in base 4, because
- 20 times m-reduction [i] would yield (121, 123, large)-net in base 4, but