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本書是英國劍橋大學卡文迪許實驗室的著名學者David J.C.MacKay博士總結多年教學經驗和科研成果,於2003年推出的一部力作。本書作者不僅透徹地論述了傳統信息論的內容和最新編碼算法,而且以高度的學科駕馭能力,匠心獨具地在一個統一框架下討論了貝葉斯數據建模、蒙特卡羅方法、聚類算法、神經網絡等屬於機器學習和推理領域的主題,從而很好地將諸多學科的技術內涵融會貫通。本書注重理論與實際的結合,內容組織科學嚴謹,反映了多門學科的內在聯係和發展趨勢。同時,本書還包含了豐富的例題和近400道習題(其中許多習題還配有詳細的解答),便於教學或自學,適合作為信息科學與技術相關專業高年級本科生和研究生教材,對相關專業技術人員也不失為一本有益的參考書。

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The Shannon capacity of a graph is a fundamental quantity in zero-error information theory measuring the rate of growth of independent sets in graph powers. Despite being well-studied, this quantity continues to hold several mysteries. Lov\'asz famously proved that the Shannon capacity of $C_5$ (the 5-cycle) is at most $\sqrt{5}$ via his theta function. This bound is achieved by a simple linear code over $\mathbb{F}_5$ mapping $x \mapsto 2x$. Motivated by this, we introduce the notion of $\textit{linear Shannon capacity}$ of graphs, which is the largest rate achievable when restricting oneself to linear codes. We give a simple proof based on the polynomial method that the linear Shannon capacity of $C_5$ is $\sqrt{5}$. Our method applies more generally to Cayley graphs over the additive group of finite fields $\mathbb{F}_q$. We compare our bound to the Lov\'asz theta function, showing that they match for self-complementary Cayley graphs (such as $C_5$), and that our bound is smaller in some cases. We also exhibit a quadratic gap between linear and general Shannon capacity for some graphs.

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The Shannon capacity of a graph is a fundamental quantity in zero-error information theory measuring the rate of growth of independent sets in graph powers. Despite being well-studied, this quantity continues to hold several mysteries. Lov\'asz famously proved that the Shannon capacity of $C_5$ (the 5-cycle) is at most $\sqrt{5}$ via his theta function. This bound is achieved by a simple linear code over $\mathbb{F}_5$ mapping $x \mapsto 2x$. Motivated by this, we introduce the notion of $\textit{linear Shannon capacity}$ of graphs, which is the largest rate achievable when restricting oneself to linear codes. We give a simple proof based on the polynomial method that the linear Shannon capacity of $C_5$ is $\sqrt{5}$. Our method applies more generally to Cayley graphs over the additive group of finite fields $\mathbb{F}_q$. We compare our bound to the Lov\'asz theta function, showing that they match for self-complementary Cayley graphs (such as $C_5$), and that our bound is smaller in some cases. We also exhibit a quadratic gap between linear and general Shannon capacity for some graphs.

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