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2025-11-04 12:43:23 -06:00
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content/CSE5313/CSE5313_L11.md
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@@ -101,7 +101,7 @@ $$
Let $\mathcal{C}=[n,k,d]_q$.
The dual code of $\mathcal{C}$ is $\mathcal{C}^\perp=\{x\in \mathbb{F}^n_q|xc^T=0\text{ for all }c\in \mathcal{C}\}$.
The dual code of $\mathcal{C}$ is $\mathcal{C}^\perp=\{x\in \mathbb{F}^n_q|xc^\top=0\text{ for all }c\in \mathcal{C}\}$.
<details>
<summary>Example</summary>
@@ -151,7 +151,7 @@ So $\langle f,h\rangle=0$.
<details>
<summary>Proof for the theorem</summary>
Recall that the dual code of $\operatorname{RM}(r,m)^\perp=\{x\in \mathbb{F}_2^m|xc^T=0\text{ for all }c\in \operatorname{RM}(r,m)\}$.
Recall that the dual code of $\operatorname{RM}(r,m)^\perp=\{x\in \mathbb{F}_2^m|xc^\top=0\text{ for all }c\in \operatorname{RM}(r,m)\}$.
So $\operatorname{RM}(m-r-1,m)\subseteq \operatorname{RM}(r,m)^\perp$.