Encoder, decoder, encoding method, and decoding method

A decoder which decodes a video, comprising:

• circuitry;
• and
• memory,
• wherein using the memory, the circuitry:
• obtains a prediction image of a current block to be decoded included in the video by one of intra prediction and inter prediction;
• decodes a transform coefficient signal of the current block;
• decodes an index value;
• selects an inverse transform basis associated with the index value from among a plurality of inverse transform bases in a common correspondence relationship between when the prediction image is obtained by intra prediction and when the prediction image is obtained by inter prediction;
• generates a prediction error signal of the current block by performing inverse transform of the transform coefficient signal using the inverse transform basis;
• and
• generates a sum of the prediction error signal and the prediction image as a reconstructed image of the current block.

An encoding method for encoding a video, comprising:

• obtaining a prediction image of a current block to be encoded included in the video by one of intra prediction and inter prediction;
• generating a difference between an image of the current block and the prediction image as a prediction error signal of the current block;
• selecting a transform basis to be used to transform the prediction error signal from among a plurality of transform bases;
• generating a transform coefficient signal of the current block by transforming the prediction error signal using the transform basis;
• encoding the transform coefficient signal;
• and
• encoding an index value associated with the transform basis among a plurality of index values associated with the plurality of transform bases in a common correspondence relationship between when the prediction image is obtained by intra prediction and when the prediction image is obtained by inter prediction.

A decoding method for decoding a video, comprising:

• obtaining a prediction image of a current block to be decoded included in the video by one of intra prediction and inter prediction;
• decoding a transform coefficient signal of the current block;
• decoding an index value;
• selecting an inverse transform basis associated with the index value from among a plurality of inverse transform bases in a common correspondence relationship between when the prediction image is obtained by intra prediction and when the prediction image is obtained by inter prediction;
• generating a prediction error signal of the current block by performing inverse transform of the transform coefficient signal using the inverse transform basis;
• and
• generating a sum of the prediction error signal and the prediction image as a reconstructed image of the current block.