Can Reed-Solomon become completely immune to wrong transmissions by using (255,1)?

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With Reed Solomon encoding, N parity bytes enable to detect up to N corrupted bytes among those received, including the parity bytes themselves. Does that mean that if we transmit 254 bytes of parity for only one byte of payload, and add a CRC check inside the byte, the transmission is virtually flawless? Is it done in applications where high datarate is of no importance?

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You can do this, but it no longer deserves to be called a Reed–Solomon code. An $(n,1)$ Reed–Solomon code is more commonly known as the $n=255$ repetition code: it repeats the single data byte $255$ times.

This is not completely immune to wrong transmissions (nothing is). However, you can detect a wrong transmission unless there is an error in every byte, and that error is the same in every byte, so it is practically immune.

Nobody actually does this.