"It might not reach."
Yuki murmured. Outside the club room window, construction noise echoed.
"What won't?" Aoi turned around.
"Isn't it difficult to have a conversation in all this noise?"
Riku laughed. "True. I keep asking people to repeat themselves."
"This is exactly what we call a noisy channel," Aoi opened the notebook.
"Channel?"
"The path connecting sender and receiver. Real channels always have noise."
Aoi drew a diagram on the whiteboard. Sender, channel, receiver. And wrote "Noise" above the channel.
"This construction sound is the noise. Our voices are the signal."
Yuki understood. "When the signal-to-noise ratio is bad, messages don't get through."
"Signal-to-noise ratio, called SNR. When it's low, communication becomes difficult."
Riku raised his hand. "Then should we just talk louder?"
"That's one method. Increase the signal to raise the SNR."
"But," Yuki thought, "is there another way?"
"Good question," Aoi smiled. "Adding redundancy."
"Redundancy?"
"Repeating the same message, or using codes that can detect and correct errors."
Aoi gave an example. "When saying 'Tomorrow, 9 o'clock, main gate,' even if parts are lost to noise, context fills in the gaps."
Riku nodded. "Even 'Tomor■ow, 9 o'clo■k, main gate' can be understood as 'Tomorrow, 9 o'clock, main gate.'"
"That's the redundancy of natural language. In information theory, we can design this more systematically."
"How?" Yuki showed interest.
"Hamming codes, Reed-Solomon codes. Add parity bits to data to detect and correct errors."
Aoi showed a simple example. "Add 3 parity bits to 4 bits of data. Single-bit errors in the 7 bits can be corrected."
"Amazing," Yuki was surprised. "But adding redundancy reduces efficiency, right?"
"That's the tradeoff. According to Shannon's channel coding theorem, as long as the rate is below channel capacity, we can communicate with arbitrarily small error rate."
"Channel capacity?"
"The maximum amount of information that can theoretically be sent through that channel. Even with noise, there's a limit."
Riku pondered. "So even in this construction noise, we can theoretically communicate perfectly?"
"With appropriate coding, yes. However, there's a cost of redundancy."
The outside noise grew even louder.
"It might not reach," Yuki said again. "But I don't want to give up."
"That's the essence of communication theory," Aoi said quietly. "Noise is unavoidable. But with clever encoding, we can deliver reliably."
Riku stood up. "My feelings are also full of noise."
"What are you talking about?" Yuki laughed.
"No, nothing," Riku blushed.
Aoi supplemented. "Human emotions also travel through noisy channels, in a sense. There are misunderstandings and miscommunications."
"So redundancy is necessary," Yuki understood. "Conveying repeatedly, expressing in different ways."
"Yes. The more important the message, the more we want to deliver it reliably. So we devise ways."
The construction noise stopped. Sudden silence.
"The noise is gone," Riku said.
"But," Aoi smiled, "we were able to converse even with the noise."
Yuki nodded. "With redundancy and the willingness to understand."
"That's humanity's strength. Optimal encoding and adaptive decoding."
Outside the window, construction noise started again. But the three's conversation didn't break. Even in noise, what you want to convey gets through. That was the hope of communication theory.