1. **Problem statement:** Find all possible binary cyclic codes of length 7, determine their minimum distance, and check if they are perfect. 2. **Background:** A binary cyclic code of length 7 is an ideal in the ring $\mathbb{F}_2[x]/(x^7-1)$. The polynomial $x^7-1$ factors over $\mathbb{F}_2$ as $$x^7-1 = (x+1)(x^3+x+1)(x^3+x^2+1).$$ 3. **Step 1: Find all divisors of $x^7-1$ in $\mathbb{F}_2[x]$.** The divisors correspond to the generator polynomials of cyclic codes. The possible generator polynomials are: - $g_1(x) = 1$ (trivial code) - $g_2(x) = x+1$ - $g_3(x) = x^3+x+1$ - $g_4(x) = x^3+x^2+1$ - $g_5(x) = (x+1)(x^3+x+1) = x^4 + x^3 + x^2 + 1$ - $g_6(x) = (x+1)(x^3+x^2+1) = x^4 + x^3 + x + 1$ - $g_7(x) = (x^3+x+1)(x^3+x^2+1) = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$ - $g_8(x) = x^7-1$ (zero code) 4. **Step 2: Determine parameters $[n,k,d]$ for each code:** - Length $n=7$ for all. - Dimension $k = n - \deg(g(x))$. | Generator | Degree | Dimension $k$ | Code Type | Minimum Distance $d$ | Perfect? | |-----------|---------|---------------|-----------|---------------------|----------| | $1$ | 0 | 7 | Whole space | 1 | No | | $x+1$ | 1 | 6 | Parity check code | 2 | No | | $x^3+x+1$| 3 | 4 | Cyclic code | 3 (Hamming code) | Yes | | $x^3+x^2+1$| 3 | 4 | Cyclic code | 3 (Hamming code) | Yes | | $x^4+x^3+x^2+1$| 4 | 3 | Dual of Hamming | 4 | No | | $x^4+x^3+x+1$| 4 | 3 | Dual of Hamming | 4 | No | | $x^6+x^5+x^4+x^3+x^2+x+1$| 6 | 1 | Repetition code | 7 | Yes | | $x^7-1$ | 7 | 0 | Zero code | Infinity (no codewords) | No | 5. **Step 3: Explanation:** - The two degree 3 polynomials generate the $(7,4,3)$ Hamming codes, which are perfect. - The repetition code $(7,1,7)$ is also perfect. - Other codes are not perfect. 6. **Final answer:** - Binary cyclic codes of length 7 correspond to the divisors of $x^7-1$ listed above. - Minimum distances and perfection as summarized in the table. Thus, the perfect binary cyclic codes of length 7 are the two $(7,4,3)$ Hamming codes and the $(7,1,7)$ repetition code.