Best Known (122−22, 122, s)-Nets in Base 4
(122−22, 122, 1492)-Net over F4 — Constructive and digital
Digital (100, 122, 1492)-net over F4, using
- 43 times duplication [i] based on digital (97, 119, 1492)-net over F4, using
- net defined by OOA [i] based on linear OOA(4119, 1492, F4, 22, 22) (dual of [(1492, 22), 32705, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(4119, 16412, F4, 22) (dual of [16412, 16293, 23]-code), using
- construction XX applied to Ce(21) ⊂ Ce(17) ⊂ Ce(16) [i] based on
- linear OA(4113, 16384, F4, 22) (dual of [16384, 16271, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(492, 16384, F4, 18) (dual of [16384, 16292, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(485, 16384, F4, 17) (dual of [16384, 16299, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(45, 27, F4, 3) (dual of [27, 22, 4]-code or 27-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(21) ⊂ Ce(17) ⊂ Ce(16) [i] based on
- OA 11-folding and stacking [i] based on linear OA(4119, 16412, F4, 22) (dual of [16412, 16293, 23]-code), using
- net defined by OOA [i] based on linear OOA(4119, 1492, F4, 22, 22) (dual of [(1492, 22), 32705, 23]-NRT-code), using
(122−22, 122, 12136)-Net over F4 — Digital
Digital (100, 122, 12136)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4122, 12136, F4, 22) (dual of [12136, 12014, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4122, 16421, F4, 22) (dual of [16421, 16299, 23]-code), using
- 2 times code embedding in larger space [i] based on linear OA(4120, 16419, F4, 22) (dual of [16419, 16299, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(4113, 16384, F4, 22) (dual of [16384, 16271, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(485, 16384, F4, 17) (dual of [16384, 16299, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(47, 35, F4, 4) (dual of [35, 28, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(4120, 16419, F4, 22) (dual of [16419, 16299, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4122, 16421, F4, 22) (dual of [16421, 16299, 23]-code), using
(122−22, 122, 7785473)-Net in Base 4 — Upper bound on s
There is no (100, 122, 7785474)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 28 269559 898752 403117 999411 115892 858552 789520 292703 132438 844891 277118 325248 > 4122 [i]