Lexicodes
Dimensions
The table below lists the dimension $k$ of the lexicode of length $n$ and minimum distance $d$, for $n \leq 63$ and $d \leq 20$. For $d > 2n/3$, the dimension is always $k = 1$. Blank cells with $d \leq n$ were not computed.
| $n \backslash d$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ | $11$ | $12$ | $13$ | $14$ | $15$ | $16$ | $17$ | $18$ | $19$ | $20$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $1$ | $1$ | |||||||||||||||||||
| $2$ | $2$ | $1$ | ||||||||||||||||||
| $3$ | $3$ | $2$ | $1$ | |||||||||||||||||
| $4$ | $4$ | $3$ | $1$ | $1$ | ||||||||||||||||
| $5$ | $5$ | $4$ | $2$ | $1$ | $1$ | |||||||||||||||
| $6$ | $6$ | $5$ | $3$ | $2$ | $1$ | $1$ | ||||||||||||||
| $7$ | $7$ | $6$ | $4$ | $3$ | $1$ | $1$ | $1$ | |||||||||||||
| $8$ | $8$ | $7$ | $4$ | $4$ | $2$ | $1$ | $1$ | $1$ | ||||||||||||
| $9$ | $9$ | $8$ | $5$ | $4$ | $2$ | $2$ | $1$ | $1$ | $1$ | |||||||||||
| $10$ | $10$ | $9$ | $6$ | $5$ | $3$ | $2$ | $1$ | $1$ | $1$ | $1$ | ||||||||||
| $11$ | $11$ | $10$ | $7$ | $6$ | $4$ | $3$ | $2$ | $1$ | $1$ | $1$ | $1$ | |||||||||
| $12$ | $12$ | $11$ | $8$ | $7$ | $4$ | $4$ | $2$ | $2$ | $1$ | $1$ | $1$ | $1$ | ||||||||
| $13$ | $13$ | $12$ | $9$ | $8$ | $5$ | $4$ | $3$ | $2$ | $1$ | $1$ | $1$ | $1$ | $1$ | |||||||
| $14$ | $14$ | $13$ | $10$ | $9$ | $6$ | $5$ | $4$ | $3$ | $2$ | $1$ | $1$ | $1$ | $1$ | $1$ | ||||||
| $15$ | $15$ | $14$ | $11$ | $10$ | $7$ | $6$ | $5$ | $4$ | $2$ | $2$ | $1$ | $1$ | $1$ | $1$ | $1$ | |||||
| $16$ | $16$ | $15$ | $11$ | $11$ | $8$ | $7$ | $5$ | $5$ | $2$ | $2$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | ||||
| $17$ | $17$ | $16$ | $12$ | $11$ | $9$ | $8$ | $6$ | $5$ | $3$ | $2$ | $2$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | |||
| $18$ | $18$ | $17$ | $13$ | $12$ | $9$ | $9$ | $7$ | $6$ | $3$ | $3$ | $2$ | $2$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | ||
| $19$ | $19$ | $18$ | $14$ | $13$ | $10$ | $9$ | $8$ | $7$ | $4$ | $3$ | $2$ | $2$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | |
| $20$ | $20$ | $19$ | $15$ | $14$ | $11$ | $10$ | $9$ | $8$ | $5$ | $4$ | $3$ | $2$ | $2$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ |
| $21$ | $21$ | $20$ | $16$ | $15$ | $12$ | $11$ | $10$ | $9$ | $5$ | $5$ | $3$ | $3$ | $2$ | $2$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ |
| $22$ | $22$ | $21$ | $17$ | $16$ | $12$ | $12$ | $11$ | $10$ | $6$ | $5$ | $4$ | $3$ | $2$ | $2$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ |
| $23$ | $23$ | $22$ | $18$ | $17$ | $13$ | $12$ | $12$ | $11$ | $6$ | $6$ | $5$ | $4$ | $2$ | $2$ | $2$ | $1$ | $1$ | $1$ | $1$ | $1$ |
| $24$ | $24$ | $23$ | $19$ | $18$ | $14$ | $13$ | $12$ | $12$ | $7$ | $6$ | $5$ | $5$ | $3$ | $2$ | $2$ | $2$ | $1$ | $1$ | $1$ | $1$ |
| $25$ | $25$ | $24$ | $20$ | $19$ | $15$ | $14$ | $12$ | $12$ | $8$ | $7$ | $6$ | $5$ | $3$ | $3$ | $2$ | $2$ | $1$ | $1$ | $1$ | $1$ |
| $26$ | $26$ | $25$ | $21$ | $20$ | $16$ | $15$ | $12$ | $12$ | $9$ | $8$ | $7$ | $6$ | $4$ | $3$ | $2$ | $2$ | $2$ | $1$ | $1$ | $1$ |
| $27$ | $27$ | $26$ | $22$ | $21$ | $17$ | $16$ | $13$ | $12$ | $9$ | $9$ | $7$ | $7$ | $5$ | $4$ | $3$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| $28$ | $28$ | $27$ | $23$ | $22$ | $18$ | $17$ | $13$ | $13$ | $10$ | $9$ | $8$ | $7$ | $5$ | $5$ | $3$ | $3$ | $2$ | $2$ | $1$ | $1$ |
| $29$ | $29$ | $28$ | $24$ | $23$ | $19$ | $18$ | $14$ | $13$ | $11$ | $10$ | $8$ | $8$ | $6$ | $5$ | $4$ | $3$ | $2$ | $2$ | $2$ | $1$ |
| $30$ | $30$ | $29$ | $25$ | $24$ | $19$ | $19$ | $15$ | $14$ | $12$ | $11$ | $9$ | $8$ | $6$ | $6$ | $5$ | $4$ | $2$ | $2$ | $2$ | $2$ |
| $31$ | $31$ | $30$ | $26$ | $25$ | $20$ | $19$ | $16$ | $15$ | $12$ | $12$ | $10$ | $9$ | $6$ | $6$ | $6$ | $5$ | $3$ | $2$ | $2$ | $2$ |
| $32$ | $32$ | $31$ | $26$ | $26$ | $21$ | $20$ | $16$ | $16$ | $13$ | $12$ | $11$ | $10$ | $7$ | $6$ | $6$ | $6$ | $3$ | |||
| $33$ | $33$ | $32$ | $27$ | $26$ | $22$ | $21$ | $17$ | $16$ | $14$ | $13$ | $11$ | $11$ | $7$ | $7$ | $6$ | $6$ | ||||
| $34$ | $34$ | $33$ | $28$ | $27$ | $23$ | $22$ | $18$ | $17$ | $14$ | $14$ | $12$ | $11$ | $8$ | $7$ | $6$ | $6$ | ||||
| $35$ | $35$ | $34$ | $29$ | $28$ | $24$ | $23$ | $19$ | $18$ | $15$ | $14$ | $13$ | $12$ | $9$ | $8$ | $7$ | $6$ | ||||
| $36$ | $36$ | $35$ | $30$ | $29$ | $25$ | $24$ | $20$ | $19$ | $16$ | $15$ | $13$ | $13$ | $10$ | $9$ | $7$ | |||||
| $37$ | $37$ | $36$ | $31$ | $30$ | $26$ | $25$ | $21$ | $20$ | $17$ | $16$ | $14$ | $13$ | $10$ | $10$ | ||||||
| $38$ | $38$ | $37$ | $32$ | $31$ | $27$ | $26$ | $22$ | $21$ | $17$ | $17$ | $15$ | $14$ | $11$ | $10$ | ||||||
| $39$ | $39$ | $38$ | $33$ | $32$ | $27$ | $27$ | $23$ | $22$ | $18$ | $17$ | $15$ | $15$ | $11$ | $11$ | ||||||
| $40$ | $40$ | $39$ | $34$ | $33$ | $28$ | $27$ | $23$ | $23$ | $19$ | $18$ | $16$ | $15$ | $12$ | $11$ | ||||||
| $41$ | $41$ | $40$ | $35$ | $34$ | $29$ | $28$ | $24$ | $23$ | $20$ | $19$ | $17$ | $16$ | $13$ | |||||||
| $42$ | $42$ | $41$ | $36$ | $35$ | $30$ | $29$ | $25$ | $24$ | $21$ | $20$ | $18$ | $17$ | $13$ | |||||||
| $43$ | $43$ | $42$ | $37$ | $36$ | $31$ | $30$ | $26$ | $25$ | $21$ | $21$ | $19$ | $18$ | ||||||||
| $44$ | $44$ | $43$ | $38$ | $37$ | $32$ | $31$ | $27$ | $26$ | $22$ | $21$ | $20$ | $19$ | ||||||||
| $45$ | $45$ | $44$ | $39$ | $38$ | $33$ | $32$ | $28$ | $27$ | $23$ | $22$ | $20$ | $20$ | ||||||||
| $46$ | $46$ | $45$ | $40$ | $39$ | $34$ | $33$ | $29$ | $28$ | $24$ | $23$ | $21$ | $20$ | ||||||||
| $47$ | $47$ | $46$ | $41$ | $40$ | $35$ | $34$ | $30$ | $29$ | $25$ | $24$ | $21$ | $21$ | ||||||||
| $48$ | $48$ | $47$ | $42$ | $41$ | $36$ | $35$ | $31$ | $30$ | $25$ | $25$ | $22$ | $21$ | ||||||||
| $49$ | $49$ | $48$ | $43$ | $42$ | $37$ | $36$ | $32$ | $31$ | $26$ | $25$ | $23$ | $22$ | ||||||||
| $50$ | $50$ | $49$ | $44$ | $43$ | $38$ | $37$ | $33$ | $32$ | $27$ | $26$ | $23$ | $23$ | ||||||||
| $51$ | $51$ | $50$ | $45$ | $44$ | $39$ | $38$ | $34$ | $33$ | $28$ | $27$ | $24$ | $23$ | ||||||||
| $52$ | $52$ | $51$ | $46$ | $45$ | $40$ | $39$ | $35$ | $34$ | $29$ | $28$ | $25$ | $24$ | ||||||||
| $53$ | $53$ | $52$ | $47$ | $46$ | $40$ | $40$ | $36$ | $35$ | $29$ | $29$ | $25$ | $25$ | ||||||||
| $54$ | $54$ | $53$ | $48$ | $47$ | $41$ | $40$ | $37$ | $36$ | $30$ | $29$ | $26$ | $25$ | ||||||||
| $55$ | $55$ | $54$ | $49$ | $48$ | $42$ | $41$ | $38$ | $37$ | $31$ | $30$ | $27$ | |||||||||
| $56$ | $56$ | $55$ | $50$ | $49$ | $43$ | $42$ | $38$ | $38$ | $32$ | $31$ | $28$ | |||||||||
| $57$ | $57$ | $56$ | $51$ | $50$ | $44$ | $43$ | $39$ | $38$ | $33$ | $32$ | $28$ | |||||||||
| $58$ | $58$ | $57$ | $52$ | $51$ | $45$ | $44$ | $40$ | $39$ | $34$ | $33$ | ||||||||||
| $59$ | $59$ | $58$ | $53$ | $52$ | $46$ | $45$ | $41$ | $40$ | $34$ | $34$ | ||||||||||
| $60$ | $60$ | $59$ | $54$ | $53$ | $47$ | $46$ | $41$ | $41$ | $35$ | $34$ | ||||||||||
| $61$ | $61$ | $60$ | $55$ | $54$ | $48$ | $47$ | $42$ | $41$ | $36$ | $35$ | ||||||||||
| $62$ | $62$ | $61$ | $56$ | $55$ | $49$ | $48$ | $43$ | $42$ | $37$ | $36$ | ||||||||||
| $63$ | $63$ | $62$ | $57$ | $56$ | $50$ | $49$ | $44$ | $43$ | $38$ | $37$ |