"It's just the two of us today."
Yuki opened the club room window. The autumn breeze felt pleasant.
"Sometimes it's good to solidify the basics slowly," Aoi said while making tea.
"Actually, there's something I still don't fully understand."
"What?"
"What exactly is channel capacity?"
Aoi sat down and opened the notebook. "Simply put, it's the maximum rate of information a channel can transmit."
"Rate?"
"Bits per unit time. For example, if a channel can send at most 100 bits per second, the capacity is 100 bps."
Yuki thought. "But with noise?"
"Sharp. With noise, capacity decreases. But it doesn't become zero."
Aoi drew a diagram.
"Sender → Channel (with noise) → Receiver"
"What Shannon proved is that at rates below capacity, error probability can be made nearly zero."
"But information gets corrupted by noise?"
"Use error correction codes. With proper design, you can reach capacity no matter how much noise there is."
Yuki was amazed. "That sounds like magic."
"Mathematical magic. However, at rates exceeding capacity, no code can avoid errors."
"The Shannon limit..."
"Yes. This is communication theory's most important theorem."
Yuki asked while drinking tea, "How do you calculate capacity?"
"Maximum mutual information. C = max I(X;Y)."
"Mutual information?"
"How much information transmission X and reception Y share. In perfect communication, I(X;Y)=H(X). With noise, it decreases."
Aoi wrote an example.
"For a binary channel with error probability p, capacity is 1 - H(p) bits."
"If p is 0, capacity is 1 bit. If p is 0.5, capacity is zero."
"With complete randomness, no information transmits," Yuki understood.
"But even if p is greater than 0.5, capacity remains positive."
"Huh? Even with more than half errors?"
"Just flip the bits. The fact that errors are frequent is itself information."
Yuki laughed. "I see! Reverse thinking."
The two continued calculating quietly.
"Aoi-senpai," Yuki suddenly said. "Information theory is like exploration."
"Exploration?"
"You don't know where the limits are. But gradually they become visible. Today I feel like I saw new scenery."
Aoi smiled. "Good metaphor. Shannon was also exploring unknown territory."
"We're following that path too."
"Yes. Equations in textbooks were once someone's great discovery."
Yuki wrote in the notebook. "Channel capacity: the width of the river where information flows."
"Good analogy."
"Noise is rocks and obstacles. But with clever navigation, water flows through."
"Error correction codes are that ingenuity."
Outside the window, a bird sang.
"Where shall we explore next?" Yuki's eyes sparkled.
"Rate-distortion theory. The dual concept of capacity."
"I'm looking forward to it."
Aoi poured more tea. "The information theory expedition has only just begun."
Their expedition continues. Deep into the channel.