"Can you hear me?"
Riku's voice was drowned out by construction noise. Schoolyard renovation had begun, and the club room was engulfed in sound.
"What? What?" Yuki shouted back.
"I said, what are we doing for club today?"
"Can't have a conversation with all this noise," Aoi chuckled wryly. "But this is a good demonstration."
"Demonstration?"
"Communication channels and noise. A core theme of information theory."
Aoi drew a diagram on the whiteboard.
"Sender → Channel → Receiver. This simple model is the foundation of all communication."
"But the channel has noise," Riku pointed at the construction sound.
"Right. Real channels always contain noise. Radio interference, cable degradation, acoustic echo. All noise."
Yuki asked, "So with noise, information can't be transmitted?"
"Not perfectly. But Shannon's channel coding theorem gives us hope."
At that moment, Mira quietly entered the room. She always appeared at the necessary moment.
"Mira, perfect timing," Aoi said. "We're talking about channel capacity."
Mira wrote in her notebook. "C = B log₂(1 + S/N)"
"Shannon's formula!" Yuki was surprised.
"Yes. C is channel capacity, B is bandwidth, S/N is signal-to-noise ratio," Aoi explained.
"Signal-to-noise ratio?"
"The ratio of signal strength to noise strength. The higher the S/N, the more information you can transmit."
Riku thought. "Then just make the signal stronger."
"That's one approach. But there are limits to power."
Mira drew another diagram. Transmitter, noise, receiver. Arrows showing information flow.
"Channel capacity is the maximum rate at which we can transmit error-free, even with noise," Aoi continued.
"Error-free? Even with noise?" Yuki asked curiously.
"That's Shannon's genius. With proper coding, at rates below channel capacity, error probability can approach zero."
Riku scratched his head. "But how?"
"Error-correcting codes. Add redundancy to data so it can be restored even if corrupted by noise."
"Redundancy means waste?"
"In communication, intentional waste is important. Hamming codes, Reed-Solomon codes, turbo codes. They all use redundancy."
Yuki wrote in her notebook. "What happens at rates above channel capacity?"
"In principle, you can't eliminate errors. No matter what code you use, errors will remain."
"Shannon limit..." Mira murmured quietly.
"Yes. A limit exists. But we can reach that limit. That's information theory's promise."
The construction noise temporarily stopped. Silence arrived.
"Like now," Aoi said. "When noise decreases, S/N increases, and channel capacity grows."
"We couldn't talk before, but now we can speak easily," Yuki realized.
Riku laughed. "Our conversation can be explained by information theory too."
"There's a concept called mutual information," Aoi continued. "The amount of information shared between sender and receiver."
Mira wrote an equation. "I(X;Y) = H(X) - H(X|Y)"
"X is the sent message, Y is the received message. H(X|Y) is conditional entropy."
"With noise, H(X|Y) increases and mutual information decreases," Aoi illustrated with a diagram.
"So less information gets through?" Yuki confirmed.
"Exactly. But we can counter with coding. That's Shannon's message."
The construction noise started again. Louder than before.
"The S/N ratio dropped!" Riku shouted, but his voice was drowned out.
Aoi used sign language. "We can use a different channel."
Yuki understood and wrote on paper. "Communicate in writing."
Mira smiled and nodded.
"There's not just one channel," Aoi wrote on paper. "Voice, text, gestures. We can combine multiple channels too."
Riku wrote on paper. "Information theory is useful!"
Amid the construction noise, the four continued conversing with paper and pen. Noise is unavoidable. But information gets through. Just as Shannon proved.