"Don't you think it's a miracle that messages arrive accurately?"
Yuki said while looking at her smartphone.
"What do you mean?" Aoi asked.
"Well, radio waves and everything are full of noise."
Aoi smiled. "Good observation. That's one of the problems information theory solved."
Mira quietly approached and wrote in her notebook. "Shannon's channel coding theorem"
"Shannon's channel coding theorem," Aoi began explaining. "Even with noisy channels, proper encoding can bring error rate arbitrarily close to zero."
"Really? Even if not perfect?"
"Perfect is impossible. But you can get arbitrarily close."
Yuki thought. "How?"
"Add redundancy," Aoi drew a diagram in her notebook. "Add check bits to the original message."
"Send extra data?"
"Yes. But that extra data allows detecting and correcting errors."
Mira wrote an example. "Hamming distance"
"Hamming distance," Aoi explained. "Measures how many bits two bit strings differ by."
"For example?"
"0000 and 0011 differ by 2 bits. Hamming distance is 2."
Yuki began to understand. "The farther the distance, the easier to distinguish?"
"Exactly. If you increase distance between codewords, even if noise changes them slightly, you can identify the original code."
Aoi continued. "For example, to correct 1-bit errors, you need codes with Hamming distance 3 or more."
"Why 3?"
"Because even if 1 bit changes, it's still closer to the original code than to other codewords."
Mira showed a calculation. "rate = k/n, k=data bits, n=total bits"
"Code rate," Aoi explained. "Encode k information bits into n total bits."
"As redundancy increases, the rate decreases?"
"Exactly. But error correction capability increases."
Yuki asked. "What's the optimal balance?"
"That's determined by channel capacity," Aoi said importantly. "What Shannon proved is that below capacity rate, errors can be made arbitrarily small."
"Above capacity?"
"No matter what encoding, errors are unavoidable."
Mira nodded quietly.
"But how do you design actual codes?" Yuki asked.
"Difficult problem. Shannon showed possibility, but concrete methods are another story."
Aoi told the history. "Hamming codes, BCH codes, Reed-Solomon codes, turbo codes, LDPC codes. Many researchers developed codes approaching the Shannon limit."
"The communication we use now too?"
"Yes. Wi-Fi, mobile phones, all use error correction codes."
Yuki was moved. "That's why messages arrive accurately."
"Not a miracle, but mathematics," Aoi smiled. "But feeling it as a miracle is important."
Mira left a note. "Perfect communication is impossible, but we can get arbitrarily close"
"Perfect communication is impossible, but we can get arbitrarily close," Yuki translated.
"Yes. This is the hope of information theory."
Yuki looked at her phone. "Even this one sentence went through multiple error corrections to arrive."
"Mathematics works in invisible places."
"The miracle of correct transmission. No, science."
Aoi nodded. "Human communication might be the same."
"What do you mean?"
"Words are imperfect. But context and expressions work as redundancy, correctly conveying intent."
Mira smiled.
"Natural error correction codes," Yuki murmured.
"Good expression."
The three nodded. The miracle of correct transmission. The everyday normal, supported by mathematics.