Best Known (109, 109+22, s)-Nets in Base 4
(109, 109+22, 5958)-Net over F4 — Constructive and digital
Digital (109, 131, 5958)-net over F4, using
- 42 times duplication [i] based on 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
(109, 109+22, 32327)-Net over F4 — Digital
Digital (109, 131, 32327)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4131, 32327, F4, 2, 22) (dual of [(32327, 2), 64523, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4131, 32773, F4, 2, 22) (dual of [(32773, 2), 65415, 23]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(4129, 32772, F4, 2, 22) (dual of [(32772, 2), 65415, 23]-NRT-code), using
- OOA 2-folding [i] 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
- OOA 2-folding [i] based on linear OA(4129, 65544, F4, 22) (dual of [65544, 65415, 23]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(4129, 32772, F4, 2, 22) (dual of [(32772, 2), 65415, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4131, 32773, F4, 2, 22) (dual of [(32773, 2), 65415, 23]-NRT-code), using
(109, 109+22, large)-Net in Base 4 — Upper bound on s
There is no (109, 131, large)-net in base 4, because
- 20 times m-reduction [i] would yield (109, 111, large)-net in base 4, but