Best Known (77, 103, s)-Nets in Base 4
(77, 103, 531)-Net over F4 — Constructive and digital
Digital (77, 103, 531)-net over F4, using
- 2 times m-reduction [i] based on digital (77, 105, 531)-net over F4, using
- trace code for nets [i] based on digital (7, 35, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- trace code for nets [i] based on digital (7, 35, 177)-net over F64, using
(77, 103, 576)-Net in Base 4 — Constructive
(77, 103, 576)-net in base 4, using
- 41 times duplication [i] based on (76, 102, 576)-net in base 4, using
- trace code for nets [i] based on (8, 34, 192)-net in base 64, using
- 1 times m-reduction [i] based on (8, 35, 192)-net in base 64, using
- base change [i] based on digital (3, 30, 192)-net over F128, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 3 and N(F) ≥ 192, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- base change [i] based on digital (3, 30, 192)-net over F128, using
- 1 times m-reduction [i] based on (8, 35, 192)-net in base 64, using
- trace code for nets [i] based on (8, 34, 192)-net in base 64, using
(77, 103, 1085)-Net over F4 — Digital
Digital (77, 103, 1085)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4103, 1085, F4, 26) (dual of [1085, 982, 27]-code), using
- 45 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 1, 5 times 0, 1, 12 times 0, 1, 21 times 0) [i] based on linear OA(497, 1034, F4, 26) (dual of [1034, 937, 27]-code), using
- construction XX applied to C1 = C([317,341]), C2 = C([319,342]), C3 = C1 + C2 = C([319,341]), and C∩ = C1 ∩ C2 = C([317,342]) [i] based on
- linear OA(491, 1023, F4, 25) (dual of [1023, 932, 26]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {317,318,…,341}, and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(491, 1023, F4, 24) (dual of [1023, 932, 25]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {319,320,…,342}, and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(496, 1023, F4, 26) (dual of [1023, 927, 27]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {317,318,…,342}, and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(486, 1023, F4, 23) (dual of [1023, 937, 24]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {319,320,…,341}, and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(41, 6, F4, 1) (dual of [6, 5, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(40, 5, F4, 0) (dual of [5, 5, 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 XX applied to C1 = C([317,341]), C2 = C([319,342]), C3 = C1 + C2 = C([319,341]), and C∩ = C1 ∩ C2 = C([317,342]) [i] based on
- 45 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 1, 5 times 0, 1, 12 times 0, 1, 21 times 0) [i] based on linear OA(497, 1034, F4, 26) (dual of [1034, 937, 27]-code), using
(77, 103, 111278)-Net in Base 4 — Upper bound on s
There is no (77, 103, 111279)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 102 847089 675156 805027 231321 161118 561066 693564 873970 673404 714302 > 4103 [i]