Best Known (21, 21+8, s)-Nets in Base 4
(21, 21+8, 257)-Net over F4 — Constructive and digital
Digital (21, 29, 257)-net over F4, using
- base reduction for projective spaces (embedding PG(7,256) in PG(28,4)) for nets [i] based on digital (0, 8, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
(21, 21+8, 258)-Net in Base 4 — Constructive
(21, 29, 258)-net in base 4, using
- 41 times duplication [i] based on (20, 28, 258)-net in base 4, using
- trace code for nets [i] based on (6, 14, 129)-net in base 16, using
- base change [i] based on digital (0, 8, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- base change [i] based on digital (0, 8, 129)-net over F128, using
- trace code for nets [i] based on (6, 14, 129)-net in base 16, using
(21, 21+8, 361)-Net over F4 — Digital
Digital (21, 29, 361)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(429, 361, F4, 8) (dual of [361, 332, 9]-code), using
- 94 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 1, 10 times 0, 1, 28 times 0, 1, 50 times 0) [i] based on linear OA(425, 263, F4, 8) (dual of [263, 238, 9]-code), using
- construction XX applied to C1 = C([254,5]), C2 = C([0,6]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C([254,6]) [i] based on
- linear OA(421, 255, F4, 7) (dual of [255, 234, 8]-code), using the primitive BCH-code C(I) with length 255 = 44−1, defining interval I = {−1,0,…,5}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(421, 255, F4, 7) (dual of [255, 234, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(425, 255, F4, 8) (dual of [255, 230, 9]-code), using the primitive BCH-code C(I) with length 255 = 44−1, defining interval I = {−1,0,…,6}, and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(417, 255, F4, 6) (dual of [255, 238, 7]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(40, 4, F4, 0) (dual of [4, 4, 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
- linear OA(40, 4, F4, 0) (dual of [4, 4, 1]-code) (see above)
- construction XX applied to C1 = C([254,5]), C2 = C([0,6]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C([254,6]) [i] based on
- 94 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 1, 10 times 0, 1, 28 times 0, 1, 50 times 0) [i] based on linear OA(425, 263, F4, 8) (dual of [263, 238, 9]-code), using
(21, 21+8, 17092)-Net in Base 4 — Upper bound on s
There is no (21, 29, 17093)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 288293 505265 488481 > 429 [i]